Horizon Therapeutics Public (HZNP) S-8 Registration of securities for employees

Solicitors 70 Sir John Rogerson’s Quay Dublin 2 Ireland D02 R296   T +353 1 232 2000 F +353 1 232 3333 W www.matheson.com DX 2 Dublin

Horizon Therapeutics plc

Connaught House, 1st Floor

1 Burlington Road

Dublin 4

Ireland

D04 C5Y6

Exhibit 5.1 to Form S-8

Our ref	30 April 2021
DF/ES 668098/1

Dear Sirs

Horizon Therapeutics Public Limited Company

Registration Statement on Form S-8

We have acted on behalf of Horizon Therapeutics Public Limited Company, a public limited company incorporated under the laws of Ireland with company number 507678 (the “Company”) in connection with the registration statement on Form S-8 (the “Registration Statement”) to be filed by the Company, pursuant to the U.S. Securities Act of 1933, as amended (the “Securities Act”), with the U.S. Securities and Exchange Commission (the “SEC”) with respect to the registration by the Company of 7,000,000 ordinary shares (nominal value of US$0.0001 per share) (the “Shares”) which may be issued and delivered under the Horizon Therapeutics Public Limited Company Amended and Restated 2020 Equity Incentive Plan (the “2020 Plan”).

1 Scope of appointment and basis of opinion

1.1 We have not investigated the laws of any country other than Ireland and this opinion is given only with respect to the laws of Ireland in effect at the date of this opinion. We have assumed, without enquiry, that there is nothing in the laws of any other jurisdiction which would or might affect the opinions as stated herein.

1.2 This opinion is to be construed in accordance with, and governed by, the laws of Ireland in force as at the date hereof, and is given solely on the basis that any issues of interpretation or liability arising hereunder may only be brought before the Irish courts, which will have exclusive jurisdiction in respect of such matters.

Managing Partner: Michael Jackson - Chairman: Tim Scanlon - Partners: Helen Kelly, Sharon Daly, Ruth Hunter, Tony O’Grady, Paraic Madigan, Tara Doyle, Anne-Marie Bohan, Patrick Spicer, Turlough Galvin, Patrick Molloy, George Brady, Robert O’Shea, Joseph Beashel, Dualta Counihan, Deirdre Dunne, Fergus Bolster, Christian Donagh, Bryan Dunne, Shane Hogan, Nicola Dunleavy, Julie Murphy-O’Connor, Mark O’Sullivan, Brian Doran, John Gill, Joe Duffy, Pat English, Shay Lydon, Aidan Fahy, Niamh Counihan, Gerry Thornton, Liam Collins, Darren Maher, Michael Byrne, Philip Lovegrove, Rebecca Ryan, Catherine O’Meara, Elizabeth Grace, Deirdre Cummins, Alan Keating, Peter McKeever, Alma Campion, Brendan Colgan, Garret Farrelly, Rhona Henry, April McClements, Gráinne Dever, Oisin McClenaghan, Rory McPhillips. Niall Pelly, Michelle Ridge, Sally-Anne Stone, Matthew Broadstock, Emma Doherty, Leonie Dunne, Stuart Kennedy, Brian McCloskey, Madeline McDonnell, Claire McLoughlin, Barry O’Connor, Donal O’Donovan, Karen Reynolds, Kevin Smith, Ciaran Healy, Chris Bollard, Deirdre Kilroy, Michael Hastings, Maria Kennedy, Barry McGettrick, Kate McKenna, Donal O’Byrne, David O’Mahony, Russell Rochford, Liam Flynn, Gráinne Callanan, Geraldine Carr, Brian Doohan, Richard Kelly, Niamh Maher, Yvonne McWeeney, Mairéad Ní Ghabháin, Padraic Roche, Vahan Tchrakian, Kieran Trant, Deirdre Crowley, Philip Tully, David Jones, Kimberley Masuda, Susanne McMenamin, David Fitzgibbon. - Tax Principals: Greg Lockhart, Catherine Galvin, Anne Harvey. - Tax Department Chair: John Ryan. - General Counsel: Dermot Powell.

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1.3 This opinion is delivered in connection with the filing of the Registration Statement with the SEC and is strictly limited to the matters stated herein and does not extend to, and is not to be read as extending, by implication or otherwise, to any other matter.

1.4 In giving this opinion, we have examined the documents and the searches made by independent law searchers on our behalf against the Company on 30 April 2021 in the Irish Companies Registration Office and in the Petitions Section and Judgements Office of the Central Office of the High Court in Dublin (the “Searches”) listed in the schedule hereto (the “Schedule”). We have also reviewed copies of such corporate records of the Company as we have deemed necessary as a basis for the opinions hereinafter expressed.

1.5 We assume no obligation to update the opinions set forth in this letter.

1.6 We have not been responsible for the investigation or verification of the facts or the reasonableness of any assumption or statement of opinion contained or represented by the Company in the Registration Statement or the 2020 Plan, nor have we attempted to determine whether any material facts have been omitted therefrom.

2 Assumptions

For the purpose of giving this opinion, we have assumed the following (without any responsibility on our part if any assumption proves to have been untrue as we have not verified independently any assumption):

2.1 the truth and accuracy of the contents of such documents and Searches reviewed by us and of any certificates of officers of the Company and / or of public officials reviewed by us;

2.2 the authenticity and completeness of all documents submitted to us as originals;

2.3 the completeness and conformity to originals of all documents supplied to us as certified, conformed or photostatic copies or received by us by facsimile or email transmission and the authenticity and completeness of the originals of such documents;

2.4 the genuineness of the signatures and seals on all original and copy documents which we have examined;

2.5 that the resolutions passed by the compensation committee of the board of directors of the Company described in the extracts referred to in the Schedule, and examined for the purposes of this opinion, were passed at a meeting of the said committee of the board of directors, that meeting was properly convened, constituted and held and that the said resolutions have not since been amended or rescinded;

2.6 that the resolution passed by the shareholders of the Company described in the Schedule, and examined for the purposes of this opinion, was passed at the annual general meeting of the Company held on 29 April 2021 (the “AGM”), that the AGM was properly convened, constituted and held and that the said resolution has not since been amended or rescinded;

2.7 that the resolutions passed by the shareholders of the Company described in the Schedule (in respect of: (i) the authority of the board of the Company to allot and issue ordinary shares for

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cash and non-cash consideration; and (ii) the authority of the board of directors of the Company to allot and issue ordinary shares for cash without first offering those ordinary shares to existing shareholders pursuant to the statutory pre-emption rights that would otherwise apply under Irish law), and examined for the purposes of this opinion, were passed at the annual general meeting of the Company held on 2 May 2019 (the “2019 AGM”), that the 2019 AGM was properly convened, constituted and held and that the said resolutions have not since been amended or rescinded;

2.8 that the offering or sale (including the marketing) of any Shares will be made, effected and conducted in accordance with and will not violate:

2.8.1 the memorandum or articles of association, from time to time, of the Company;

2.8.2 any applicable laws and regulations (including, without limitation, (i) the securities laws and regulations of any jurisdiction (including Ireland) or supra-national authority which impose any restrictions, or mandatory requirements, in relation to the offering or sale of any shares to the public in any jurisdiction (including Ireland) and any prospectus (or analogous disclosure document) prepared in connection therewith; and (ii) the competition, anti-trust or merger control laws and regulations of any jurisdiction (including Ireland) or supra-national authority); and

2.8.3 any requirement or restriction imposed by any court, governmental body or supra-national authority having jurisdiction over the Company or the members of its group;

2.9 that the memorandum and articles of association of the Company reviewed by us are the current memorandum and articles of association, are up to date and have not been amended or superseded;

2.10 that the information disclosed by the Searches was accurate at the date the Searches were made and has not been altered and that the Searches did not fail to disclose any information which had been delivered for registration but did not appear from the information available at the time the Searches were made or which ought to have been delivered for registration at that time but had not been so delivered and that no additional matters would have been disclosed by Searches carried out since that time;

2.11 that:

2.11.1 the Company was, or will be, fully solvent (i) at the time of, and immediately after, the filing of the Registration Statement and (ii) at the date hereof;

2.11.2 the Company would not, as a consequence of doing any act or thing which the Registration Statement contemplates, permits or requires the Company to do, be insolvent; and

2.11.3 no steps have been taken or, to the best of the knowledge, information and belief of the directors of the Company, are being taken to appoint a receiver, liquidator or an examiner over the Company or any part of its undertaking or assets, or to strike the Company off the Register of Companies or to otherwise dissolve or wind up the Company;

2.12 that the Registration Statement becomes and remains effective;

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2.13 that the Shares are, upon exercise of awards granted under the 2020 Plan, duly paid for (including payment of the par value) and allotted and issued in accordance with the terms of the 2020 Plan; and

2.14 the truth of all representations and information given to us in reply to any queries we have made which we have considered necessary for the purposes of giving this opinion.

3 Opinion

Based upon and subject to the foregoing, to the qualifications set out in this letter and to any matter not disclosed to us, we are of the opinion that, so far as the laws of Ireland are concerned, the Company has sufficient authorised but unissued ordinary shares free from statutory pre-emption rights of shareholders to satisfy full exercise of awards granted under the 2020 Plan were such exercise to occur today and, upon issuance in accordance with the 2020 Plan and the Registration Statement, and subject to receipt by the Company of the full consideration payable in respect thereof, the Shares will be legally issued, fully paid and non-assessable (which term means that no shareholder shall be obliged to pay or contribute further amounts to the capital of the Company in connection with the issue of such shares).

4 Qualifications

The opinions set forth in this opinion are given subject to the following qualifications:

4.1 a search of the Companies Registration Office (the “CRO”) is not capable of revealing whether or not a winding up petition or a petition for the appointment of an examiner has been presented; and

4.2 a search at the Petitions Section of the Central Office of the High Court is not capable of revealing whether or not a receiver has been appointed.

Whilst each of the making of a winding up order, the making of an order for the appointment of an examiner and the appointment of a receiver may be revealed by a search at the CRO, it may not be filed at the CRO immediately and, therefore, our searches at the CRO may not have revealed such matters.

5 Consent

We hereby consent to the filing of this opinion as an exhibit to the Registration Statement and to the use of our name therein. In giving this consent, we do not hereby admit that we come within the category of persons whose consent is required under section 7 of the Securities Act or the rules and regulations of the SEC promulgated thereunder.

This opinion is furnished to you and the persons entitled to rely upon it pursuant to the applicable provisions of the Securities Act for use in connection with the filing of the Registration Statement and is not to be used, circulated, quoted or otherwise referred to for any other purpose without our express written consent.

Yours faithfully

/s/ Matheson

MATHESON

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Schedule 1

Documents examined for the purpose of giving this opinion

1. A copy of the 2020 Plan.

2. The final form of the Registration Statement to be filed by the Company with the SEC.

3. A certificate executed by a director or the company secretary which:

(a) annexes:

(i) the certificate of incorporation of the Company;

(ii) the certificates of incorporation of the Company on change of name; and

(iii) the memorandum and articles of association of the Company in force as at the date of certificate; and

(b) annexes a copy extract of the resolutions approved by:

(i) the Compensation Committee of the Board of Directors of the Company on 17 February 2021; and

(ii) shareholders of the Company passed at the AGM; and

(c) annexes a copy extract of the relevant resolutions approved by shareholders of the Company passed at the 2019 AGM; and

(d) confirms that the number of authorised but unissued ordinary shares in the capital of the Company exceeds the aggregate of: (i) the number of ordinary shares that would be issued on exercise of outstanding awards granted under the 2020 Plan were such exercise to occur today; and (ii) the maximum number of ordinary shares that are capable of being issued on conversion of all convertible securities, exercise of any warrants, exercise of any options or rights to be allotted shares or exercise or discharge of any analogous securities or rights.

4. The results of Searches.

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