Globally catch exceptions in a WPF application?

AppDomain.UnhandledException Event

This event provides notification of uncaught exceptions. It allows the application to log information about the exception before the system default handler reports the exception to the user and terminates the application.

   public App()
   {
      AppDomain currentDomain = AppDomain.CurrentDomain;
      currentDomain.UnhandledException += new UnhandledExceptionEventHandler(MyHandler);    
   }

   static void MyHandler(object sender, UnhandledExceptionEventArgs args) 
   {
      Exception e = (Exception) args.ExceptionObject;
      Console.WriteLine("MyHandler caught : " + e.Message);
      Console.WriteLine("Runtime terminating: {0}", args.IsTerminating);
   }

If the UnhandledException event is handled in the default application domain, it is raised there for any unhandled exception in any thread, no matter what application domain the thread started in. If the thread started in an application domain that has an event handler for UnhandledException, the event is raised in that application domain. If that application domain is not the default application domain, and there is also an event handler in the default application domain, the event is raised in both application domains.

For example, suppose a thread starts in application domain "AD1", calls a method in application domain "AD2", and from there calls a method in application domain "AD3", where it throws an exception. The first application domain in which the UnhandledException event can be raised is "AD1". If that application domain is not the default application domain, the event can also be raised in the default application domain.

In addition what others mentioned here, note that combining the Application.DispatcherUnhandledException (and its similars) with

<configuration>
  <runtime>  
    <legacyUnhandledExceptionPolicy enabled="1" />
  </runtime>
</configuration>

in the app.config will prevent your secondary threads exception from shutting down the application.

Example code using NLog that will catch exceptions thrown from all threads in the AppDomain, from the UI dispatcher thread and from the async functions:

App.xaml.cs :

public partial class App : Application
{
    private static Logger _logger = LogManager.GetCurrentClassLogger();

    protected override void OnStartup(StartupEventArgs e)
    {
        base.OnStartup(e);

        SetupExceptionHandling();
    }

    private void SetupExceptionHandling()
    {
        AppDomain.CurrentDomain.UnhandledException += (s, e) =>
            LogUnhandledException((Exception)e.ExceptionObject, "AppDomain.CurrentDomain.UnhandledException");

        DispatcherUnhandledException += (s, e) =>
        {
            LogUnhandledException(e.Exception, "Application.Current.DispatcherUnhandledException");
            e.Handled = true;
        };

        TaskScheduler.UnobservedTaskException += (s, e) =>
        {
            LogUnhandledException(e.Exception, "TaskScheduler.UnobservedTaskException");
            e.SetObserved();
        };
    }

    private void LogUnhandledException(Exception exception, string source)
    {
        string message = $"Unhandled exception ({source})";
        try
        {
            System.Reflection.AssemblyName assemblyName = System.Reflection.Assembly.GetExecutingAssembly().GetName();
            message = string.Format("Unhandled exception in {0} v{1}", assemblyName.Name, assemblyName.Version);
        }
        catch (Exception ex)
        {
            _logger.Error(ex, "Exception in LogUnhandledException");
        }
        finally
        {
            _logger.Error(exception, message);
        }
    }

Use the Application.DispatcherUnhandledException Event. See this question for a summary (see Drew Noakes' answer).

Be aware that there'll be still exceptions which preclude a successful resuming of your application, like after a stack overflow, exhausted memory, or lost network connectivity while you're trying to save to the database.