We aim at constructing a theory of sheaves and sheaf cohomology on Chu spaces in the framework of the theory of Grothendieck sites. We will also study the structure on the normal Chu spaces defined by Grothendieck topologies arising on them in a natural way and then apply the results to information systems and event structures. This book involves methods of homological algebra, sheaf theory on Grothendieck sites, and the theory of sheaf cohomology of partially ordered sets. We prove that the Grothendieck and Cech cohomologies of Chu spaces are isomorphic. We give a characterization of the cohomological dimension and the dual dimension of a normal Chu space. We develop a theory of flabby sheaves and flabby dimensions for normal Chu spaces. We prove the acyclicity of flabby sheaves.