Error handling and logging are essential aspects of building robust and maintainable Web APIs in ASP.NET Core. You can implement error handling by using exception filters, middleware, or a centralized error handling middleware like UseExceptionHandler. By handling exceptions, you can return appropriate error responses with meaningful information to the clients. Additionally, you can integrate a logging framework like Serilog or NLog to capture and log errors, warnings, and information about the application's behavior. This helps in troubleshooting and monitoring the application's health and performance.