applications $\Sigma$ will usually be compact but this will not be important in the following. Because hyperbolic problems can be localized we can focus our attention on some neighbourhood of a given point $p \in \Sigma$. We shall mainly be concerned with evolutions local in time that cover some neighbourhood of ${\cal S}$ (which includes a neighbourhood of $\Sigma$ in ${\cal T}$). Problems arising in long term evolutions will be addressed only if they shed light on the need to control geometric uniqueness.

Given an $ST$-vacuum-space-time, one may ask [*which data need to be given on ${\cal S}$ and ${\cal T}$ to reconstruct the space-time uniquely by solving the field equations*]{}. A first answer to this question is given in [@friedrich:nagy], where it is shown that Einstein’s field equations admit well-posed initial boundary value problems. One of the main difficulties overcome in this work is to ensure that gauge conditions and/or constraints are preserved. For reasons discussed below this task is more subtle here than in the case of the Cauchy problem.

In the subsequent years there was a certain activity concerned with ‘constraint preserving’ formulations of initial boundary value problems in numerical relativity (cf. [@alcubierre] for some of the 