  Let (S,H) be a polarized K3 surface. We define Brill-Noether filtration on
moduli spaces of vector bundles on S. Assume that (c_1(E),H) > 0 for a sheaf E
in the moduli space. We give a formula for the expected dimension of the
Brill-Noether subschemes. Following the classical theory for curves, we give a
notion of Brill-Noether generic K3 surfaces.
  Studying correspondences between moduli spaces of sheaves of different ranks
on S, we prove our main theorem: polarized K3 surface which is generic in sense
of moduli is also generic in sense of Brill-Noether theory (here H is the
positive generator of the Picard group of S). In case of algebraic curves such
a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is
sometimes called ``the strong theorem of the Brill-Noether theory''.
  We finish by considering a number of projective examples. In particular, we
construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and
show the relation with the theory of Brill-Noether special curves.
