Kestrel shutdown function in Startup.cs in ASP.NET Core

In ASP.NET Core you can register to the cancellation tokens provided by IApplicationLifetime

public class Startup 
{
    public void Configure(IApplicationBuilder app, IApplicationLifetime applicationLifetime) 
    {
        applicationLifetime.ApplicationStopping.Register(OnShutdown);
    }

    private void OnShutdown()
    {
         // Do your cleanup here
    }
}

IApplicationLifetime is also exposing cancellation tokens for ApplicationStopped and ApplicationStarted as well as a StopApplication() method to stop the application.

For .NET Core 3.0+

From comments @Horkrine

For .NET Core 3.0+ it is recommended to use IHostApplicationLifetime instead, as IApplicationLifetime will be deprecated soon. The rest will still work as written above with the new service

In addition to the original answer, I had an error while trying to wire the IApplicationLifetime within the constructor.

I solved this by doing:

public class Startup 
{
    public void Configure(IApplicationBuilder app) 
    {
        var applicationLifetime = app.ApplicationServices.GetRequiredService<IApplicationLifetime>();
        applicationLifetime.ApplicationStopping.Register(OnShutdown);
    }

    private void OnShutdown()
    {
         // Do your cleanup here
    }
}

This class is now obsolete, please refer to the new interface IHostApplicationLifetime. More info here.