  Given a stratified variety X with strata satisfying a cohomological
parity-vanishing condition, we define and show the uniqueness of "parity
sheaves", which are objects in the constructible derived category of sheaves
with coefficients in an arbitrary field or complete discrete valuation ring.
This construction depends on the choice of a parity function on the strata.
  If X admits a resolution also satisfying a parity condition, then the direct
image of the constant sheaf decomposes as a direct sum of parity sheaves, and
the multiplicities of the indecomposable summands are encoded in certain
refined intersection forms appearing in the work of de Cataldo and Migliorini.
We give a criterion for the Decomposition Theorem to hold in the semi-small
case.
  Our framework applies to many stratified varieties arising in representation
theory such as generalised flag varieties, toric varieties, and nilpotent
cones. Moreover, parity sheaves often correspond to interesting objects in
representation theory. For example, on flag varieties we recover in a unified
way several well-known complexes of sheaves. For one choice of parity function
we obtain the indecomposable tilting perverse sheaves. For another, when using
coefficients of characteristic zero, we recover the intersection cohomology
sheaves and in arbitrary characteristic the special sheaves of Soergel, which
are used by Fiebig in his proof of Lusztig's conjecture.
