There are two ways you can do this, dependent on the version of the language you are using.

C# 5.0 and above

You can use the async and await keywords to simplify a great deal of this for you.

async and await were introduced into the language to simplify using the Task Parallel Library, preventing you from having to use ContinueWith and allowing you to continue to program in a top-down manner.

Because of this, you can simply use a ABC0/catch block to catch the exception, like so:

try
{
// Start the task.
var task = Task.Factory.StartNew<StateObject>(() => { /* action */ });


// Await the task.
await task;
}
catch (Exception e)
{
// Perform cleanup here.
}

Note that the method encapsulating the above must use have the async keyword applied so you can use await.

C# 4.0 and below

You can handle exceptions using the ContinueWith overload that takes a value from the TaskContinuationOptions enumeration, like so:

// Get the task.
var task = Task.Factory.StartNew<StateObject>(() => { /* action */ });


// For error handling.
task.ContinueWith(t => { /* error handling */ }, context,
TaskContinuationOptions.OnlyOnFaulted);

The OnlyOnFaulted member of the TaskContinuationOptions enumeration indicates that the continuation should only be executed if the antecedent task threw an exception.

Of course, you can have more than one call to ContinueWith off the same antecedent, handling the non-exceptional case:

// Get the task.
var task = new Task<StateObject>(() => { /* action */ });


// For error handling.
task.ContinueWith(t => { /* error handling */ }, context,
TaskContinuationOptions.OnlyOnFaulted);


// If it succeeded.
task.ContinueWith(t => { /* on success */ }, context,
TaskContinuationOptions.OnlyOnRanToCompletion);


// Run task.
task.Start();

You can create some custom Task factory, which will produce Tasks with exception handling processing embedded. Something like this:

using System;
using System.Threading.Tasks;


class FaFTaskFactory
{
public static Task StartNew(Action action)
{
return Task.Factory.StartNew(action).ContinueWith(
c =>
{
AggregateException exception = c.Exception;


// Your Exception Handling Code
},
TaskContinuationOptions.OnlyOnFaulted | TaskContinuationOptions.ExecuteSynchronously
).ContinueWith(
c =>
{
// Your task accomplishing Code
},
TaskContinuationOptions.OnlyOnRanToCompletion | TaskContinuationOptions.ExecuteSynchronously
);
}


public static Task StartNew(Action action, Action<Task> exception_handler, Action<Task> completion_handler)
{
return Task.Factory.StartNew(action).ContinueWith(
exception_handler,
TaskContinuationOptions.OnlyOnFaulted | TaskContinuationOptions.ExecuteSynchronously
).ContinueWith(
completion_handler,
TaskContinuationOptions.OnlyOnRanToCompletion | TaskContinuationOptions.ExecuteSynchronously
);
}
};

You can forget about exceptions processing for Tasks produced from this factory in your client code. In the same time you still can wait finishing of such Tasks or use them in Fire-and-Forget style:

var task1 = FaFTaskFactory.StartNew( () => { throw new NullReferenceException(); } );
var task2 = FaFTaskFactory.StartNew( () => { throw new NullReferenceException(); },
c => {    Console.WriteLine("Exception!"); },
c => {    Console.WriteLine("Success!"  ); } );


task1.Wait(); // You can omit this
task2.Wait(); // You can omit this

But if be honest I'm not really sure why you want to have completion handling code. In any case this decision depends on the logic of your application.