  This paper investigates the Witt groups of triangulated categories of sheaves
(of modules over a ring R in which 2 is invertible) equipped with
Poincare-Verdier duality. We consider two main cases, that of perfect complexes
of sheaves on locally compact Hausdorff spaces and that of cohomologically
constructible complexes of sheaves on polyhedra. We show that the Witt groups
of the latter form a generalised homology theory for polyhedra and continuous
maps. Under certain restrictions on the ring R, we identify the constructible
Witt groups of a finite simplicial complex with Ranicki's free symmetric
L-groups. Witt spaces are the natural class of spaces for which the rational
intersection homology groups have Poincare duality. When the ring R is the
rationals we show that every Witt space has a natural L-theory, or Witt,
orientation and we identify the constructible Witt groups with the 4-periodic
colimit of the bordism groups of Witt spaces. This allows us to interpret
Goresky and Macpherson's L-classes of singular spaces as stable homology
operations from the constructible Witt groups to rational homology.
