Direct dial: +44 20 7614 2224 Email: craja@cgsh.com Exhibit 5.1 26 July 2023 LivaNova PLC 20 Eastbourne Terrace London W2 6LG Re: LivaNova PLC – Registration Statement on Form S-8 Ladies and Gentlemen: We have acted as special English counsel to LivaNova PLC (the “Company”), a public limited company incorporated under the laws of England and Wales, in connection with the preparation and filing with the Securities and Exchange Commission (the “Commission”) under the Securities Act of 1933, as amended (the “Securities Act”), of a registration statement on Form S-8 (such registration statement, including the documents incorporated by reference therein, the “Registration Statement”) for the registration of an additional: (i) 650,000 ordinary shares with a nominal value of £1 each which may be issued or transferred under the Amended and Restated LivaNova PLC 2022 Incentive Award Plan (the “A&R Plan”); and (ii) 650,000 ordinary shares with a nominal value of £1 each that may become available for issuance under the A&R Plan, as a result of outstanding awards under the A&R Plan being either (a) forfeited, lapsed, or expired, (b) converted to shares of another company in connection with a takeover, sale, recapitalisation, reorganisation, merger, consolidation, split-up, spin-off, combination, exchange of shares or other similar event, (c) settled for cash (in whole or in part), or (d) repurchased by the Company ((i) and (ii), together, being the “Ordinary Shares”). In connection with the Registration Statement, we have been asked to provide an opinion on certain matters as set out below. We have taken instruction in this regard solely from the Company. We understand that the Ordinary Shares are not and are not intended to be admitted to trading on any market or exchange, or otherwise listed, in the United Kingdom. In arriving at the opinions expressed below, we have reviewed the following documents: (a) the Registration Statement; and

2 (b) a certificate dated 26 July 2023 signed by an authorised signatory of the Company (the “Company’s Certificate”) relating to certain factual matters and having annexed thereto and certified as true, complete and up-to-date copies of the following documents: (i) the current articles of association of the Company adopted with effect from 29 June 2020 (the “Articles”); (ii) the resolutions passed by the Company’s Board of Directors (the “Board”) on 19 April 2023 (the “Board Resolutions”); and (iii) the ordinary resolution passed by the shareholders of the Company at the Company’s annual general meeting held on 12 June 2023 (the “2023 AGM”) approving the A&R Plan (the “Shareholders’ Resolution”). In addition, we have reviewed the originals, or copies certified or otherwise identified to our satisfaction, of all such other documents, and we have made such investigations of law, as we have deemed appropriate as a basis for the opinions expressed below. In rendering the opinions expressed below we have assumed and not verified: (a) the genuineness of all signatures, stamps and seals, the authenticity and completeness of all documents supplied to us and the conformity to the originals of all documents supplied to us as photocopies, facsimile or electronic copies; (b) that, where a document has been examined by us in draft, specimen or certificated form, it has been or will be executed in the form of that draft, specimen or certificate; (c) the accuracy as to factual matters of each document we have reviewed, including, without limitation, the accuracy and completeness of all statements in the Company’s Certificate; (d) that the Company has fully complied with its obligations under all applicable anti-terrorism, anti-money laundering, sanctions and human rights legislation, and that each allotment and issue of Ordinary Shares in the manner contemplated in the Registration Statement will be compliant with such laws; (e) that no document has been entered into by any of the parties thereto in connection with any money laundering or any other unlawful activity; (f) that all consents, approvals, notices, filings and registrations which are necessary under any applicable laws or regulations (other than laws or regulations of England and Wales) in order to permit the allotment and issue of any Ordinary Shares in the manner contemplated in the Registration Statement have been or will be duly made or obtained;

3 (g) that there are no provisions of the laws of any jurisdiction outside England and Wales that would have any implication for the opinions we express and that, insofar as the laws of any jurisdiction outside England and Wales may be relevant to this opinion letter, such laws have been and will be complied with; (h) that the Company has complied with all applicable provisions of the Financial Services Act 2012 and the Financial Services and Markets Act 2000, as amended (the “FSMA”) and any applicable secondary legislation made under any of the foregoing with respect to anything done by the Company in relation to the Ordinary Shares from or otherwise involving the United Kingdom (including Sections 19 (carrying on a regulated activity), 21 (financial promotion), and 85 (public offers) of FSMA); (i) that at the time of each allotment and issue of any Ordinary Shares, the Company shall have received in full consideration equal to the subscription price or other consideration for such Ordinary Shares and shall have entered the holder or holders thereof in the register of members of the Company showing that all such Ordinary Shares shall have been fully paid up as to their nominal value and any premium thereon as at each allotment date; (j) that at the time of each issue of any Ordinary Share, the issue price or other consideration for such issue is not less than the nominal value, if any, of the Ordinary Share; (k) that the meeting of the Board held on 19 April 2023 at which the Board Resolutions were passed was duly convened and held and such resolutions are a true record of the proceedings at such meeting and are valid, in full force and effect and have not been amended, revoked or superseded; (l) that the 2023 AGM at which the Shareholders’ Resolution was passed was duly convened and held and the Shareholders’ Resolution was duly and validly passed at the 2023 AGM, is in full force and effect and has not been amended, revoked or superseded; (m) that each director of the Company has disclosed any interest which he or she may have in the transactions contemplated by the Board Resolutions in accordance with the provisions of the Companies Act 2006 and the Articles, and that none of the relevant directors of the Company have any interest in such transactions except to the extent permitted by the Articles; and (n) that the directors consider in good faith that the actions to be carried out pursuant to the Board Resolutions by the Company will promote the success of the Company for the benefit of its members as a whole. Based on the foregoing, and subject to the further qualifications and limitations set forth below, it is our opinion that the Ordinary Shares, when issued in

4 accordance with all necessary corporate action of the Company, the provisions of the Company’s articles of association in force at that time and the A&R Plan, and subject to receipt by the Company of the aggregate issue price or other consideration in respect of such Ordinary Shares, will be validly issued, fully paid and non-assessable (for the purposes of this opinion, “non-assessable” means that no further contributions in respect of the Ordinary Shares will be required to be made to the Company by the holders thereof, by reason solely of their being such holders). The opinion set out above is limited to the laws of England and Wales in force as at the date of this opinion letter (taking into account the effect of the Agreement on the withdrawal of the United Kingdom of Great Britain and Northern Ireland from the European Union and the European Atomic Energy Community and the European Union (Withdrawal) Act 2018), as currently applied by the courts in England and Wales, and are given on the basis that this opinion letter and any non-contractual obligations arising out of or in connection with it will be governed by and construed in accordance with English law. We hereby consent to the use of this opinion as a part Exhibit 5.1 of the Registration Statement as counsel for the Company who have passed on the validity of the Ordinary Shares being registered by the Registration Statement. In giving such consent, we do not thereby admit that we are within the category of persons whose consent is required under Section 7 of the Securities Act or the rules and regulations of the Commission thereunder. The opinion expressed herein is rendered on and as of the date hereof, and we assume no obligation to advise you or any other person, or to make any investigations, as to any legal developments or factual matters arising subsequent to the date hereof that might affect the opinions expressed herein. Very truly yours, CLEARY GOTTLIEB STEEN & HAMILTON LLP By: __________________ _ Chrishan Raja, a Partner