The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular.
The concept was introduced in the work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a consequence of the Riemann-Hilbert correspondence, which establishes a connection between the derived categories regular holonomic D-modules and constructible sheaves. Perverse sheaves are the objects in the latter that correspond to individual D-modules (and not more general complexes thereof); a perverse sheaf is in general represented by a complex of sheaves. The concept of perverse sheaves is already implicit in a 75's paper of Kashiwara on the constructibility of solutions of holonomic D-modules.
A key observation was that the intersection homology of Mark Goresky and Robert MacPherson could be described using sheaf complexes that are actually perverse sheaves. It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role in number theory, algebra, and representation theory.