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.