  This is a chapter in an upcoming book on aperiodic order. We go over
different versions of tiling cohomology (\v Cech, pattern-equivariant, PV,
quotient) with emphasis on the inverse limit constructions used to compute
these cohomologies. We then consider the uses of tiling cohomology to
distinguish spaces, to understand deformations, and to help understand maps
between tiling spaces. The emphasis of this chapter is on substitution tilings
and their generalizations, but the underlying ideas apply equally well to
cut-and-project tilings and to tilings defined by local matching rules.
