In this Thesis I investigate how Fano manifolds equipped with a Kahler-
Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff
topology) and some of the relations of this question with Algebraic Geometry.
A central topic in the Thesis is the study of the deformation theory for
singular Kahler-Einstein metrics. In particular, it is shown that Kahler-
Einstein Fano varieties of dimension two (Del Pezzo surfaces) with only
nodes as singularities and discrete automorphism group, admit (partial)
smoothings which also carry (orbifold) Kahler-Einstein metrics. The above
result is then used to study the metric compactification in the Gromov-
Hausdorff topology of the space of Kahler-Einstein Del Pezzo surfaces. In
the case of cubic surfaces some evidence is provided that the metric compactification
agrees with the classical algebraic compactification given by
the set of Chow polystable cubics.
Finally, I study some higher dimensional analogous of the results outlined
above: for example, we briefly discuss the existence and deformation
theory for K¨ahler-Einstein metrics on nodal Fano varieties and the compactifications
of the space of intersections of two quadrics in P5.