The theory of derivators is an approach to homotopical algebra that focuses on the idea of enhancing the classical homotopy category $ho(C)$ of a homotopy theory $C$ by the collection of the homotopy categories of diagrams in $C$ all at once. The resulting objects turn out to be much richer than the homotopy category alone and this viewpoint has been useful for expressing homotopical universal properties. At the same time, this approach is different from (and less strong than) the methods of higher category theory - which has been developed and used in recent years for related purposes with great impact in various areas of research. I will survey the basic theory, applications and examples of derivators, and then I will discuss the general notion of an $\infty$-derivator, as a natural higher categorical extension of ordinary derivators. This generalization is based on the use of the homotopy $n$-category, for $1 \leq n \leq \infty$, it bridges the gap between derivators and $\infty$-categories, and it provides a common framework of reference for both types of objects/approaches.