Is it possible to override a non-virtual method?

No, you cannot override a non-virtual method. The closest thing you can do is hide the method by creating a new method with the same name but this is not advisable as it breaks good design principles.

But even hiding a method won't give you execution time polymorphic dispatch of method calls like a true virtual method call would. Consider this example:

using System;

class Example
{
    static void Main()
    {
        Foo f = new Foo();
        f.M();

        Foo b = new Bar();
        b.M();
    }
}

class Foo
{
    public void M()
    {
        Console.WriteLine("Foo.M");
    }
}

class Bar : Foo
{
    public new void M()
    {
        Console.WriteLine("Bar.M");
    }
}

In this example both calls to the M method print Foo.M. As you can see this approach does allow you to have a new implementation for a method as long as the reference to that object is of the correct derived type but hiding a base method does break polymorphism.

I would recommend that you do not hide base methods in this manner.

I tend to side with those who favor C#'s default behavior that methods are non-virtual by default (as opposed to Java). I would go even further and say that classes should also be sealed by default. Inheritance is hard to design for properly and the fact that there is a method that is not marked to be virtual indicates that the author of that method never intended for the method to be overridden.

Edit: "execution time polymorphic dispatch":

What I mean by this is the default behavior that happens at execution time when you call virtual methods. Let's say for example that in my previous code example, rather than defining a non-virtual method, I did in fact define a virtual method and a true overridden method as well.

If I were to call b.Foo in that case, the CLR would correctly determine the type of object that the b reference points to as Bar and would dispatch the call to M appropriately.

You can't override non-virtual method of any class in C# (without hacking CLR), but you can override any method of interface the class implements. Consider we have non-sealed

class GraphicsDevice: IGraphicsDevice {
    public void DoWork() {
        Console.WriteLine("GraphicsDevice.DoWork()");
    }
}

// with its interface
interface IGraphicsDevice {
    void DoWork();
}

// You can't just override DoWork in a child class,
// but if you replace usage of GraphicsDevice to IGraphicsDevice,
// then you can override this method (and, actually, the whole interface).

class MyDevice: GraphicsDevice, IGraphicsDevice {
    public new void DoWork() {
        Console.WriteLine("MyDevice.DoWork()");
        base.DoWork();
    }
}

And here's demo

class Program {
    static void Main(string[] args) {

        IGraphicsDevice real = new GraphicsDevice();
        var myObj = new MyDevice();

        // demo that interface override works
        GraphicsDevice myCastedToBase = myObj;
        IGraphicsDevice my = myCastedToBase;

        // obvious
        Console.WriteLine("Using real GraphicsDevice:");
        real.DoWork();

        // override
        Console.WriteLine("Using overriden GraphicsDevice:");
        my.DoWork();

    }
}

No you can't.

You can only override a virtual method - see the MSDN here:

In C#, derived classes can contain methods with the same name as base class methods.

• The base class method must be defined virtual.

If the base class isn't sealed then you can inherit from it and write a new method that hides the base one (use the "new" keyword in the method declaration). Otherwise no, you cannot override it because it was never the original authors intent for it to be overridden, hence why it isn't virtual.