Versioning in ASP.NET Core Web API allows you to handle changes to your API's contract while maintaining backward compatibility. You can implement versioning by adding a versioning middleware, such as Microsoft.AspNetCore.Mvc.Versioning, and configuring it in the ConfigureServices method of the Startup class. Versioning can be done through URL-based versioning, query string-based versioning, or header-based versioning. This enables clients to specify the desired version of the API they want to interact with.