  These are lecture notes based on a series of talks given by the authors at
the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held
in Istanbul in Summer of 2005. They provide an introduction to a recent work on
the complex ball uniformization of the moduli spaces of Del Pezzo surfaces, K3
surfaces and algebraic curves of lower genus. We discuss the relationship of
these constructions with the Deligne-Mostow theory of periods of hypergeometric
differentianl forms. For convenience to a non-expert reader we include an
introduction to the theory of periods of integrals on algebraic varieties with
emphasis on abelian varieties and K3 surfaces.
