Logging and error handling are important aspects of building robust and maintainable Web APIs. In ASP.NET Core Web API, you can implement logging by using the built-in ILogger interface and logging providers like Serilog, NLog, or the default logging provider. Logging captures important information, warnings, and errors, enabling troubleshooting and monitoring of the API's behavior. Error handling can be implemented using middleware, exception filters, or the UseExceptionHandler middleware. By capturing exceptions and returning appropriate error responses, you can handle errors gracefully and provide meaningful feedback to clients.