  This work consists of two parts. In the first part we develop new techniques
to compute Koszul cohomology groups for several classes of varieties. As
applications we prove results on projective normality and syzygies for
algebraic surfaces. From more general results we obtain in particular the
following:
  a) Mukai's conjecture (and stronger variants of it) regarding projective
normality and normal presentation for surfaces with Kodaira dimension 0, and
uniform bounds for higher syzygies associated to adjoint linear series,
  b) effective bounds along the lines of Mukai's conjecture regarding
projective normality and normal presentation for surfaces of positive Kodaira
dimension, and,
  c) results on projective normality for pluricanonical models of surfaces of
general type (recovering and strengthening results by Ciliberto) and
generalizations of them to higher syzygies.
  In the second part we prove results on very ampleness, projective normality
and higher syzygies for both singular and smooth Calabi-Yau threefolds. The
general results that we prove are analogues of the results of St. Donat for K3
surfaces. From these general results we obtain bounds very close to Fujita's
conjecture regarding very ampleness of powers of an ample line bundle A (for
instance, if A^3 is greater than 1, our bound is one short of Fujita's). We
generalize our results on very ampleness and projective normality to higher
syzygies.
