Exhibit 5.1

June 10, 2020

BioCryst Pharmaceuticals, Inc.

4505 Emperor Blvd., Suite 200

Durham, North Carolina 27703

Re: BioCryst Pharmaceuticals, Inc. Registration Statement on Form S-8

Ladies and Gentlemen:

We have examined the Registration Statement on Form S-8 (the “Registration Statement”) of BioCryst Pharmaceuticals, Inc., a Delaware corporation (the “Company”), filed with the Securities and Exchange Commission (the “Commission”) pursuant to the Securities Act of 1933, as amended (the “Securities Act”), in connection with the offering by the Company of up to 1,700,000 shares of the Company’s common stock, par value $0.01 per share (“Common Stock”), issuable pursuant to the Company’s Inducement Equity Incentive Plan (as amended and restated February 7, 2020, the “Plan,” and such shares, the “Shares”).

In arriving at the opinion expressed below, we have examined originals, or copies certified or otherwise identified to our satisfaction as being true and complete copies of the originals, of specimen Common Stock certificates and such other documents, corporate records, certificates of officers of the Company and of public officials and other instruments as we have deemed necessary or advisable to enable us to render this opinion.  In our examination, we have assumed without independent investigation the genuineness of all signatures, the legal capacity and competency of all natural persons, the authenticity of all documents submitted to us as originals and the conformity to original documents of all documents submitted to us as copies.  As to any facts material to this opinion, we have relied to the extent we deemed appropriate and without independent investigation upon statements and representations of officers and other representatives of the Company and others.  We have also assumed without independent investigation that there are no agreements or understandings between or among the Company and any participants in the Plan that would expand, modify, or otherwise affect the terms of the Plan or the respective rights or obligations of the participants thereunder.

Based on the foregoing and in reliance thereon, and subject to the assumptions, exceptions, qualifications and limitations set forth herein, we are of the opinion that the Shares, when issued against payment therefor in accordance with the terms set forth in the Plan, will be validly issued, fully paid and non-assessable.

BioCryst Pharmaceuticals, Inc.

June 10, 2020

Page 2

The opinion expressed above is subject to the following exceptions, qualifications, limitations and assumptions:

A.          We render no opinion herein as to matters involving the laws of any jurisdiction other than the Delaware General Corporation Law (“DGCL”). This opinion is limited to the effect of the current state of the DGCL and the facts as they currently exist. We assume no obligation to revise or supplement this opinion in the event of future changes in such laws or the interpretation thereof or such facts.  We express no opinion regarding any state securities laws or regulations.

We consent to the filing of this opinion as an exhibit to the Registration Statement.  In giving this 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.

Very truly yours,

/s/ Gibson, Dunn & Crutcher LLP