1 Introduction

General Relativity (GR) is a peculiar physical theory. GR is about geometries, and not fields, in a given space-time. That is, the theory’s solutions are not metric tensors – or possible other matter tensors – but rather the equivalence class of these tensors under arbitrary smooth relabeling of points in space-time. This peculiarity makes the task of analyzing the dynamics of the theory difficult. One is used to evolving tensor fields, in fact tensor components, in a given coordinate system; while here the extra freedom of the theory makes these components non-unique. The values of some of these components can be given arbitrarily. Only certain relations between them are invariant – and so have a physical meaning. In particular, some components can be made arbitrarily large and rough, while the geometry is, for instance, flat. Thus, it is often hard to see, from just comparing tensor components, whether two solutions, that is two geometries, are close to each other during evolution. To overcome this problem, several proposals have been made to fix the evolution in a unique way and at the same time obtain well behaved solutions. In general, these proposals provide for equation systems equivalent, in a sense to be discussed at length later, to Einstein’s equations which are hyperbolic, that is, whose evolution is continuous as a function of the initial data. This property is vital for many applications, ranging from Newtonian approximations to numerical simulations. The aim of this work is to review these proposals, paying special attention to the applications where they have proven fruitful.