Task continuation on UI thread

Call the continuation with TaskScheduler.FromCurrentSynchronizationContext():

    Task UITask= task.ContinueWith(() =>
    {
     this.TextBlock1.Text = "Complete"; 
    }, TaskScheduler.FromCurrentSynchronizationContext());

This is suitable only if the current execution context is on the UI thread.

With async you just do:

await Task.Run(() => do some stuff);
// continue doing stuff on the same context as before.
// while it is the default it is nice to be explicit about it with:
await Task.Run(() => do some stuff).ConfigureAwait(true);

However:

await Task.Run(() => do some stuff).ConfigureAwait(false);
// continue doing stuff on the same thread as the task finished on.

If you have a return value you need to send to the UI you can use the generic version like this:

This is being called from an MVVM ViewModel in my case.

var updateManifest = Task<ShippingManifest>.Run(() =>
    {
        Thread.Sleep(5000);  // prove it's really working!

        // GenerateManifest calls service and returns 'ShippingManifest' object 
        return GenerateManifest();  
    })

    .ContinueWith(manifest =>
    {
        // MVVM property
        this.ShippingManifest = manifest.Result;

        // or if you are not using MVVM...
        // txtShippingManifest.Text = manifest.Result.ToString();    

        System.Diagnostics.Debug.WriteLine("UI manifest updated - " + DateTime.Now);

    }, TaskScheduler.FromCurrentSynchronizationContext());