Do you have to put Task.Run in a method to make it async?

When you use Task.Run to run a method, Task gets a thread from threadpool to run that method. So from the UI thread's perspective, it is "asynchronous" as it doesn't block UI thread.This is fine for desktop application as you usually don't need many threads to take care of user interactions.

However, for web application each request is serviced by a thread-pool thread and thus the number of active requests can be increased by saving such threads. Frequently using threadpool threads to simulate async operation is not scalable for web applications.

True Async doesn't necessarily involving using a thread for I/O operations, such as file / DB access etc. You can read this to understand why I/O operation doesn't need threads. http://blog.stephencleary.com/2013/11/there-is-no-thread.html

In your simple example,it is a pure CPU-bound calculation, so using Task.Run is fine.

One of the most important thing to remember when decorating a method with async is that at least there is one await operator inside the method. In your example, I would translate it as shown below using TaskCompletionSource.

private Task<int> DoWorkAsync()
{
    //create a task completion source
    //the type of the result value must be the same
    //as the type in the returning Task
    TaskCompletionSource<int> tcs = new TaskCompletionSource<int>();
    Task.Run(() =>
    {
        int result = 1 + 2;
        //set the result to TaskCompletionSource
        tcs.SetResult(result);
    });
    //return the Task
    return tcs.Task;
}

private async Task DoWork()
{
    int result = await DoWorkAsync();
}

First, let's clear up some terminology: "asynchronous" (async) means that it may yield control back to the calling thread before it starts. In an async method, those "yield" points are await expressions.

This is very different than the term "asynchronous", as (mis)used by the MSDN documentation for years to mean "executes on a background thread".

To futher confuse the issue, async is very different than "awaitable"; there are some async methods whose return types are not awaitable, and many methods returning awaitable types that are not async.

Enough about what they aren't; here's what they are:

• The async keyword allows an asynchronous method (that is, it allows await expressions). async methods may return Task, Task<T>, or (if you must) void.
• Any type that follows a certain pattern can be awaitable. The most common awaitable types are Task and Task<T>.

So, if we reformulate your question to "how can I run an operation on a background thread in a way that it's awaitable", the answer is to use Task.Run:

private Task<int> DoWorkAsync() // No async because the method does not need await
{
  return Task.Run(() =>
  {
    return 1 + 2;
  });
}

(But this pattern is a poor approach; see below).

But if your question is "how do I create an async method that can yield back to its caller instead of blocking", the answer is to declare the method async and use await for its "yielding" points:

private async Task<int> GetWebPageHtmlSizeAsync()
{
  var client = new HttpClient();
  var html = await client.GetAsync("http://www.example.com/");
  return html.Length;
}

So, the basic pattern of things is to have async code depend on "awaitables" in its await expressions. These "awaitables" can be other async methods or just regular methods returning awaitables. Regular methods returning Task/Task<T> can use Task.Run to execute code on a background thread, or (more commonly) they can use TaskCompletionSource<T> or one of its shortcuts (TaskFactory.FromAsync, Task.FromResult, etc). I don't recommend wrapping an entire method in Task.Run; synchronous methods should have synchronous signatures, and it should be left up to the consumer whether it should be wrapped in a Task.Run:

private int DoWork()
{
  return 1 + 2;
}

private void MoreSynchronousProcessing()
{
  // Execute it directly (synchronously), since we are also a synchronous method.
  var result = DoWork();
  ...
}

private async Task DoVariousThingsFromTheUIThreadAsync()
{
  // I have a bunch of async work to do, and I am executed on the UI thread.
  var result = await Task.Run(() => DoWork());
  ...
}

I have an async/await intro on my blog; at the end are some good followup resources. The MSDN docs for async are unusually good, too.