Как я могу использовать Assert, чтобы проверить, что было сгенерировано исключение?

Check out nUnit Docs for examples about:

[ExpectedException( typeof( ArgumentException ) )]

This is going to depend on what test framework are you using?

In MbUnit, for example, you can specify the expected exception with an attribute to ensure that you are getting the exception you really expect.

[ExpectedException(typeof(ArgumentException))]

It is an attribute on the test method... you don't use Assert. Looks like this:

[ExpectedException(typeof(ExceptionType))]
public void YourMethod_should_throw_exception()

Be wary of using ExpectedException, as it can lead to several pitfalls as demonstrated here:

http://geekswithblogs.net/sdorman/archive/2009/01/17/unit-testing-and-expected-exceptions.aspx

And here:

http://xunit.github.io/docs/comparisons.html

If you need to test for exceptions, there are less frowned upon ways. You can use the try{act/fail}catch{assert} method, which can be useful for frameworks that don't have direct support for exception tests other than ExpectedException.

A better alternative is to use xUnit.NET, which is a very modern, forward looking, and extensible unit testing framework that has learned from all the others mistakes, and improved. One such improvement is Assert.Throws, which provides a much better syntax for asserting exceptions.

You can find xUnit.NET at github: http://xunit.github.io/

If you're using MSTest, which originally didn't have an ExpectedException attribute, you could do this:

try 
{
    SomeExceptionThrowingMethod()
    Assert.Fail("no exception thrown");
}
catch (Exception ex)
{
    Assert.IsTrue(ex is SpecificExceptionType);
}

Usually your testing framework will have an answer for this. But if it's not flexible enough, you can always do this:

try {
    somethingThatShouldThrowAnException();
    Assert.Fail(); // If it gets to this line, no exception was thrown
} catch (GoodException) { }

As @Jonas points out, this DOES NOT work for catching a base Exception:

try {
    somethingThatShouldThrowAnException();
    Assert.Fail(); // raises AssertionException
} catch (Exception) {
    // Catches the assertion exception, and the test passes
}

If you absolutely must catch Exception, you need to rethrow the Assert.Fail(). But really, this is a sign you shouldn't be hand-writing this; check your test framework for options, or see if you can throw a more meaningful exception to test for.

catch (AssertionException) { throw; }

You should be able to adapt this approach to whatever you like -- including specifying what kinds of exceptions to catch. If you only expect certain types, finish the catch blocks off with:

} catch (GoodException) {
} catch (Exception) {
    // not the right kind of exception
    Assert.Fail();
}