Exhibit 5.1

Commerce Court West, 199 Bay Street

P.O. Box 247, Suite 4405

Toronto, Ontario

Canada M5L 1E8

+1.416.360.8484

November 30, 2018

The Board of Directors

of The Bank of Nova Scotia

44 King Street West

Scotia Plaza, 8th Floor

Toronto, Ontario M5H 1H1

Canada

The Bank of Nova Scotia

Registration Statement on Form F-3

Ladies and Gentlemen:

We have acted as United States counsel to The Bank of Nova Scotia, a Canadian bank chartered under the Bank Act (Canada) (the “Bank”), in connection with Registration Statement on Form F-3 (such registration statement and any amendments thereto, the “Registration Statement”) filed by the Bank with the Securities and Exchange Commission (the “Commission”) under the Securities Act of 1933, as amended (the “Securities Act”), relating to the offering from time to time, as set forth in the prospectus contained in the Registration Statement (the “Prospectus”) and as to be set forth in one or more supplements to the Prospectus (each a “Prospectus Supplement”), of: (a) the Bank’s common shares, without par value (the “Common Shares”); (b) the Bank’s preferred shares, without par value (the “Preferred Shares” and, together with the Common Shares, the “Shares”); (c) the Bank’s senior debt securities, in one or more series (the “Senior Debt Securities”) and (d) the Bank’s subordinated debt securities, in one or more series (the “Subordinated Debt Securities” and, together with the Senior Debt Securities, the “Debt Securities” and the Debt Securities, together with the Shares, the “Securities”).

The Senior Debt Securities may be issued from time to time in one or more series, with each series to be issued by the Bank pursuant to the senior debt securities indenture dated as of January 22, 2010 (the “Senior Debt Indenture”) among the Bank, Computershare Trust Company, N.A., as U.S. trustee (the “U.S. Trustee”), and Computershare Trust Company of Canada, as Canadian trustee (the “Canadian Trustee” and, together with the U.S. Trustee, the “Trustees”), as amended and supplemented by a first supplemental indenture thereto dated as of November 30, 2018, among the Bank and the Trustees (the “First Supplemental Senior Debt Indenture”), and as such indenture may be further amended or supplemented for any series of Senior Debt Securities. The Subordinated Debt Securities may be issued from time to time in one or more series, with each series to be issued by the Bank pursuant to either (i) the subordinated debt securities indenture dated as of December 16, 2015, among the Bank and the Trustees (the “Subordinated Debt Indenture”) or (ii) the subordinated debt securities indenture (Non-Viability Contingent Capital (NVCC)) dated as of October 12, 2017, among the Bank and the Trustees (the

SHEARMAN.COM

Shearman & Sterling LLP is a limited liability partnership organized in the United States under the laws of the state of Delaware, which laws limit the personal liability of partners.

Page  2

“NVCC Debt Indenture” and, together with the Senior Debt Indenture, as amended by the First Supplemental Senior Debt Indenture, and the Subordinated Debt Indenture, the “Indentures”), as such indentures may be amended or supplemented for any series of Subordinated Debt Securities. The Senior Debt Indenture, the Subordinated Debt Indenture, the NVCC Debt Indenture and the First Supplemental Senior Debt Indenture are filed as Exhibits 4.1, 4.2, 4.3 and 4.4 to the Registration Statement, respectively. The Securities may be sold from time to time pursuant to certain underwriting agreements (the “Underwriting Agreements”) among the Bank and the various underwriters named therein in the forms filed as exhibits to the Registration Statement or in such forms as may be filed in connection with the offering of such Securities.

In that connection, we have reviewed originals or copies of the following documents:

(a) the Registration Statement;

(b) the Prospectus;

(c) the forms of the Underwriting Agreements;

(d) the Senior Debt Indenture;

(e) the Subordinated Debt Indenture;

(f) the NVCC Debt Indenture;

(g) the First Supplemental Senior Debt Indenture; and

(h) the forms of certificates representing the Debt Securities.

The documents described in the foregoing clauses (a) through (h) are collectively referred to herein as the “Opinion Documents.”

We have also reviewed originals or copies of such other corporate records of the Bank, certificates of public officials and of officers of the Bank and agreements and other documents as we have deemed necessary as a basis for the opinions expressed below.

In our review of the Opinion Documents and the other documents, we have assumed:

(a) The genuineness of all signatures.

(b) The authenticity of the originals of the documents submitted to us.

(c) The conformity to authentic originals of any documents submitted to us as copies.

(d) As to matters of fact, the truthfulness of the representations made in the Opinion Documents and in certificates of public officials and officers of the Bank.

Page  3

(e) That each of the Opinion Documents is or will be the legal, valid and binding obligation of each party thereto, other than the Bank, enforceable against each such party in accordance with its terms.

(f) That:

(1) The Bank is an entity duly organized and validly existing under the laws of the jurisdiction of its organization.

(2) The Bank has power and authority (corporate or otherwise) to execute, deliver and perform, and has duly authorized, executed and delivered or will duly authorize, execute and deliver (except to the extent Generally Applicable Law (as defined below) is applicable to such execution and delivery), the Opinion Documents to which it is a party.

(3) The execution, delivery and performance by the Bank of the Opinion Documents to which it is a party do not and will not:

(i) contravene its charter, by-laws or other organizational documents; or

(ii) except with respect to Generally Applicable Law, violate any law, rule or regulation applicable to it.

(g) The execution, delivery and performance by the Bank of the Opinion Documents to which it is a party do not and will not result in any conflict with or breach of any agreement or document binding on it.

(h) That, except with respect to Generally Applicable Law, no authorization, approval, consent or other action by, and no notice to or filing with, any governmental authority or regulatory body or any other third party is required for the due execution, delivery or performance by the Bank of any Opinion Document to which it is a party or, if any such authorization, approval, consent, action, notice or filing is required, it has been or will be, prior to execution of such Opinion Document, duly obtained, taken, given or made and is or will be, prior to execution of such Opinion Document, in full force and effect.

(i) That the Registration Statement shall have become effective and such effectiveness shall not have been terminated or rescinded.

We are attorneys admitted to practice in the State of New York and we do not express any opinion as to the laws of any other jurisdiction other than the federal law of the United States of America and, to the extent the laws of the Province of Ontario or the laws of Canada may be relevant to this opinion letter, we have, with your permission, assumed the correctness of the opinion letter of your local counsel with respect to Ontario and Canadian law addressed to you and dated the date hereof, without independently investigating or verifying the matters covered thereby. Our opinions are subject to the assumptions and limitations set forth in such opinions with respect thereto.

We have not independently established the validity of the foregoing assumptions.

Page  4

“Generally Applicable Law” means the federal law of the United States of America, and the law of the State of New York (including in each case the rules or regulations promulgated thereunder or pursuant thereto), that a New York lawyer exercising customary professional diligence would reasonably be expected to recognize as being applicable to the Bank, the Opinion Documents or the transactions governed by the Opinion Documents. Without limiting the generality of the foregoing definition of Generally Applicable Law, the term “Generally Applicable Law” does not include any law, rule or regulation that is applicable to the Bank, the Opinion Documents or such transactions solely because such law, rule or regulation is part of a regulatory regime applicable to any party to any of the Opinion Documents or any of its affiliates due to the specific assets or business of such party or such affiliate.

Based upon the foregoing and upon such other investigation as we have deemed necessary and subject to the assumptions and qualifications set forth herein, we are of the opinion that:

1. The Senior Debt Indenture has been duly executed and delivered by the Bank, to the extent such execution and delivery is a matter of New York law, and is the legal, valid and binding obligation of the Bank, enforceable against the Bank in accordance with its terms.

2. The Subordinated Debt Indenture has been duly executed and delivered by the Bank, to the extent such execution and delivery is a matter of New York law, and is the legal, valid and binding obligation of the Bank, enforceable against the Bank in accordance with its terms.

3. The NVCC Debt Indenture has been duly executed and delivered by the Bank, to the extent such execution and delivery is a matter of New York law, and is the legal, valid and binding obligation of the Bank, enforceable against the Bank in accordance with its terms.

4. The First Supplemental Senior Debt Indenture has been duly executed and delivered by the Bank, to the extent such execution and delivery is a matter of New York law, and is the legal, valid and binding obligation of the Bank, enforceable against the Bank in accordance with its terms.

5. When the Debt Securities have been duly executed by the Company, authenticated by the Trustee in accordance with the applicable Indenture and delivered and paid for as provided in the applicable Underwriting Agreement, the Debt Securities will be the legal, valid and binding obligations of the Bank, enforceable against the Bank in accordance with their terms and entitled to the benefits of the applicable Indenture.

Our opinions expressed above are subject to the following qualifications:

(a) (i) The effect of any applicable bankruptcy, insolvency, reorganization, moratorium or similar laws affecting creditors’ rights generally (including without limitation all laws relating to fraudulent transfers) and (ii) possible judicial action giving effect to governmental actions or foreign laws affecting creditors’ rights.

(b) The effect of general principles of equity, including without limitation concepts of materiality, reasonableness, good faith and fair dealing (regardless of whether considered in a proceeding in equity or at law).

Page  5

(c) Our opinions are limited to Generally Applicable Law and we do not express any opinion herein concerning any other law.

(d) Although the Indentures provide for obligations of the Company denominated in a currency other than United States dollars, we express no opinion as to whether a court would award a judgment in a currency other than United States dollars.

(e) We express no opinion with respect to the enforceability of any indemnity against any loss in converting into a specified currency the proceeds or amount of a court judgment in another currency.

(f) We express no opinion as to the validity, binding effect or enforceability of Section 301(b) and Section 1601(a) of the Senior Debt Indenture, as supplemented by the First Supplemental Senior Debt Indenture (and the corresponding provisions of the Senior Debt Securities issued thereunder), of Section 301(b) or Article Fifteen of the Subordinated Debt Indenture (and the corresponding provisions of the Subordinated Debt Securities issued thereunder) or of Section 301(b) or Article Twelve of the NVCC Debt Indenture (and the corresponding provisions of the Subordinated Debt Securities issued thereunder), each of which is governed by the laws of the Province of Ontario and the laws of Canada.

(g) We express no opinion with respect to Section 16(a) of the Underwriting Agreements, Section 1501 of the Senior Debt Indenture, Section 1601 of the Subordinated Debt Indenture or Section 1401 of the NVCC Debt Indenture to the extent that such Sections contain a waiver of any objection based on inappropriate venue or forum non conveniens in any federal court of the United States.

This opinion letter speaks only as of the date hereof. We expressly disclaim any responsibility to advise you of any development or circumstance of any kind, including any change of law or fact, that may occur after the date of this opinion letter and which might affect the opinions expressed herein.

We hereby consent to the filing of this opinion letter as an exhibit to the Registration Statement dated the date hereof filed by the Bank and to the use of our name under the heading “Legal Matters” in the Prospectus. In giving this consent, we do not hereby admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act or the rules and regulations of the Commission promulgated thereunder.

Very truly yours,

/s/ Shearman & Sterling LLP

JRL/KJR/JDE/EB

LN