How do 3D engines typically handle object rotation? (Illustrated)

I'm building a quick 3D engine for a game I'm developing.

Currently I'm handling rotation of 3D objects by rotating around the global axis Y, X then Z. Is this the correct way to do it, or should I be rotating objects about its local axis?

Which do you recommend and why?

If the local axis method is the correct way, would a 3D 3x3 or 4x4 matrix have to be used to calculate the rotations correctly? How?

Solution

4x4 matrices give you more flexibility. For example, you can concatenate several transformations (translation, rotation, etc) through matrix multiplication.