  The aim of this survey is to give an overview on the geometry of Einstein
maximal globally hyperbolic 2+1 spacetimes of arbitrary curvature, conatining a
complete Cauchy surface of finite type. In particular a specialization to the
finite type case of the canonicla Wick rotation-rescaling theory, previously
developed by the authors, is provided. This includes, for arbitrary curvatures,
parameterizations in terms of suitable measured geodesic laminations on open
hyperbolic surfaces of finite type. The same geometric objects also
parameterize complex projective structures on the surfaces. The coincidence of
such parameter space is explained by means of geometric correlations between
spacetimes of different curvatures and projective surfaces realized via
canonical WR-rescaling along the cosmological times. We also specialize on AdS
case mostly referring to recent results achieved by other authors. In
particular we describe maximal causal extensions of AdS globally hyperbolic
spacetimes and an AdS approach to the theory of earthquakes for hyperbolic
surfaces of finite type. A general earthquake theorem is proved for the so
called enhanced Teichmuller space. The case of spacetimes with conical timelike
singularities is also treated.
