EX-5.1

Exhibit 5.1

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Our ref		6 August 2020
PR 668098/1

Dear Sirs

We act as Irish legal advisers to Horizon Therapeutics Public Limited Company (the “Company”) who have asked us to render this Opinion as to matters of the law of Ireland in connection with the offering by the Company of up to 13,570,000 new ordinary shares (“Ordinary Shares”), including up to 1,770,000 Ordinary Shares that may be sold pursuant to the exercise of an option to purchase additional shares, pursuant to a Registration Statement on Form S-3 (the “Registration Statement”) filed by the Company under the Securities Act of 1933, as amended (the “Securities Act”), the related prospectus dated 4 March 2019 included in the Registration Statement (the “Base Prospectus”), the prospectus supplement relating to the Shares dated 6 August 2020 filed with the Securities Exchange Commission pursuant to Rule 424(b) promulgated under the Securities Act (the “Prospectus Supplement”) and, together with the Base Prospectus, the (“Prospectus”) and an underwriting agreement dated 6 August 2020 (the “Underwriting Agreement”) between the Company and the Underwriters (as defined therein).

We consent to the reference to our firm under the caption “Legal Matters” in the Prospectus Supplement, the filing of this opinion as an exhibit to a Current Report on Form 8-K and the incorporation by reference of this opinion in the Registration Statement.

1	Limit to Opinion

This Opinion is strictly limited to the matters expressly stated below and no statements or opinions may be inferred beyond such matters.

1.1	Laws of Ireland

This Opinion is based upon and limited to the law of Ireland as is in effect on the date hereof and is based on legislation published and cases fully reported before that date. We have assumed without enquiry that there is nothing in the law of any other jurisdiction which would or might affect our opinions as stated herein. For the avoidance of doubt, the law of Ireland does not include the law in force in Northern Ireland.

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1.2	Laws of Other Jurisdictions

We have made no investigations of and we express no opinion in this Opinion on the law of any other jurisdiction or the effect thereof.

1.3	Update Opinion

We assume no obligation to update the opinions set forth in this Opinion and we assume no obligation to inform any Addressee of any change in law (including any change in interpretation of law).

1.4	Reliance

We have relied, without any investigation with respect to factual matters, solely upon the representations made by the Company in the Underwriting Agreement and the Certificate (as defined in Schedule 1).

2	Assumptions

The opinions in this Opinion have been made on the following assumptions:

2.1	Genuine

All signatures, initials, seals and stamps contained in or on the Certificate (and the attachments thereto) are genuine.

2.2	Authentic and Complete

Any document submitted to us as a copy (including without limitation any document submitted to us as a pdf (or any other format)) or attachment to an email conforms to the original of such document and all original documents are authentic and complete with all requisite seals and stamps affixed.

2.3	Certificate

The Certificate is accurate in all respects (other than in relation to matters on which we expressly opine).

2.4	Searches

The information disclosed by the Searches (as defined in Schedule 1) was accurate and complete as of the date the Searches were made and has not been altered. The Searches did not fail to disclose any information which had been delivered for registration but which 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. No additional matters would have been disclosed by searches being carried out since that time.

2.5	Corporate Documents

The copies of minutes of meetings and / or of resolutions reviewed by us correctly record the proceedings at such meetings and the subject matter which they purport to record and that any meetings referred to in such copies were duly convened, duly quorate and held, that those present at any such meetings were entitled to attend and vote at the meeting and acted bona fide throughout and that no further resolutions have been passed or other action taken which would or might alter the effectiveness thereof.

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2.6	Authority

None of the resolutions and authorities of the shareholders or directors of the Company upon which we have relied have been varied, amended or revoked in any respect or have expired.

3	Opinion

Based upon and subject to the assumptions set out above, we are of the opinion that:

3.1	Due Incorporation

The Company is duly incorporated in Ireland for an indefinite period as a separate legal entity and is validly existing under the law of Ireland and is subject to suit in its own name.

3.2	Issuance of Shares

The Ordinary Shares, when issued and paid for in accordance with the terms of the Underwriting Agreement, will be validly issued, fully paid, and not subject to calls for any additional payments.

This opinion is intended solely for your benefit and is not to be made available to or to be relied upon by any other person, firm, or entity without our prior written consent.

Yours faithfully

/s/ Matheson

MATHESON

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Schedule 1

Documents and Searches

1	Documents

For the purpose of this Opinion, we have examined only the following documents (pdf copies of which were received by us):

1.1	A certificate issued by the secretary of the Company dated 6 August 2020 (the “Certificate”).

2	Searches

We have arranged only for the following searches to be carried out by independent law searchers on 6 August 2020 in respect of the Company:

2.1

Judgments Office search of the Central Office of the High Court of Ireland;

2.2

2.3

Companies Registration Office search to find out:

(i)

if an examiner or liquidator has been appointed to the Company; and

(ii)

if the Company has been dissolved.

(together, the “Searches”).