The only known sufficient condition for
the existence of a Kahler-Einstein metric on a Fano manifold
can be formulated in terms of so-called alpha-invariant
introduced by Tian and Yau more than 20 years ago.
This invariant can be defined for any log Fano variety with
log terminal singularities using purely algebraic language.
Using a global-to-local results of Shokurov, one can define
a similar invariants for a germ of log terminal singularity.
We describe the role played by the alpha-invariant
in birational geometry and singularity theory.
We prove the existence of Kahler-Einstein metrics on
many quasismooth well-formed weighted del Pezzo hypersurface
and compare this result with new obstructions found
by J.Gauntlett, D.Martelli, J.Sparks and S.-T.Yau.
We apply our technique to classify weakly-exceptional
quasismooth well-formed weighted del Pezzo hypersurface
using the classification of isolated rational quasihomogeneous
three-dimensional singularities obtained by S.S.T.Yau and Y.Yu.