  We introduce complex singularity exponents of plurisubharmonic functions and
prove a general semi-continuity result for them. This concept contains as a
special case several similar concepts which have been considered e.g. by Arnold
and Varchenko, mostly for the study of hypersurface singularities. The
plurisubharmonic version is somehow based on a reduction to the algebraic case,
but it also takes into account more quantitative informations of great interest
for complex analysis and complex differential geometry. We give as an
application a new derivation of criteria for the existence of K\"ahler-Einstein
metrics on certain Fano orbifolds, following Nadel's original ideas (but with a
drastic simplication in the technique, once the semi-continuity result is taken
for granted). In this way, 3 new examples of rigid K\"ahler-Einstein Del Pezzo
surfaces with quotient singularities are obtained.
