Exhibit 5.1

March 3, 2021                                    

Texas Capital Bancshares, Inc.,

2000 McKinney Avenue, Suite 700,

Dallas, Texas 75201

Ladies and Gentlemen:

We are acting as counsel to Texas Capital Bancshares, Inc., a Delaware corporation (the “Company”), in connection with the registration under the Securities Act of 1933 (the “Act”) of 12,000,000 depositary shares (the “Depositary Shares”), each representing a 1/40th interest in a share of the Company’s Fixed Rate Non-Cumulative Perpetual Preferred Stock, Series B, par value of $0.01 per share, and having a liquidation preference of $1,000 per share (the “Preferred Shares”). The Depositary Shares are being issued pursuant to the Deposit Agreement, dated February March 3, 2021 (the “Deposit Agreement”), between the Company, Computershare Trust Company, N.A. and Computershare, Inc. as joint depositary (the “Depositary”), and the holders from time to time of the depositary receipts issued pursuant to the Deposit Agreement (the “Depositary Receipts”). The Depositary Shares are evidenced by the Depositary Receipts.

We have examined such corporate records, certificates and other documents, and such questions of law, as we have considered necessary or appropriate for the purposes of this opinion. Upon the basis of such examination, we advise you that, in our opinion, the Preferred Shares have been validly issued and are fully paid and nonassessable, and the Depositary Receipts evidencing the Depositary Shares have been validly issued and entitle the holders thereof to the rights specified in the Depositary Receipts and the Deposit Agreement, subject to bankruptcy, insolvency, fraudulent transfer, reorganization, moratorium and similar laws of general applicability relating to or affecting creditors’ rights and to general equity principles.

In rendering the foregoing opinion, we are not passing upon, and assume no responsibility for, any disclosure in any registration statement or any related prospectus or other offering material relating to the offer and sale of the Depository Shares.

The foregoing opinion is limited to the Federal laws of the United States, the laws of the State of New York and the General Corporation Law of the State of Delaware, and we are expressing no opinion as to the effect of the laws of any other jurisdiction.

We have relied as to certain factual matters on information obtained from public officials, officers of the Company and other sources believed by us to be responsible, and we have assumed that the Preferred Shares have been deposited with the Depositary in accordance with the Deposit Agreement, that the Deposit Agreement has been duly authorized, executed and delivered by the Depositary, that the certificates evidencing the Preferred Shares and the Depositary Receipts conform to the specimens thereof examined by us, that the Depositary Receipts have been duly executed and delivered by one of the Depositary’s authorized officers and, if necessary, have been duly countersigned by the registrar for the Depositary Receipts, that the certificate evidencing the Preferred Shares has been duly countersigned and registered by the registrar and transfer agent of the Preferred Shares and that the signatures on all documents examined by us are genuine, assumptions which we have not independently verified.

We hereby consent to the filing of this opinion as an exhibit to a Current Report on Form 8-K to be incorporated by reference into the Registration Statement relating to the Depositary Shares and the Preferred Shares and to references to us under the heading “Validity of the Securities” in the Prospectus Supplement relating to the Depositary Shares and the Preferred Shares, dated February 24, 2021. In giving such consent, we do not thereby admit that we are in the category of persons whose consent is required under Section 7 of the Act.

Very truly yours,

/s/ Sullivan & Cromwell LLP