Why do quantum computers outperform classical ones? Is this due to the presence of entanglement? interference? non-commutativity? ... Understanding the relation between quantum and classical computation is a great challenge, both of fundamental and practical importance. In the field of classical simulations, the goal is to identify quantum computations that can be simulated efficiently on a classical computer and to understand which ingredients are responsible for the existence of such simulations. In this talk we discuss a selection of standard results and our own work in this topic. This includes e.g. classical simulations of certain highly entangled computations, the surprising power of commuting quantum operations and the existence of classical simulations of the quantum Fourier transform.