In this chapter, we introduce several very general objects, which will be used to interpret the results of Hodge theory concerning the de Rham cohomology of a Kähler manifold, and to apply them from a more theoretical and conceptual point of view. First, we need to introduce the notion of a sheaf (of abelian groups, rings, modules, etc.) over a topological space X. A sheaf F is the following collection of data: a group (ring, module, etc.)F(U) of sections of F on U, for each open set U of X, together with restriction maps F(U) → F(V) for V ⊂ U. We require that a section of F on U is determined by its restrictions to the open sets V of a covering of U, and conversely, that a section can be constructed by gluing together sections of the open sets of a covering, under the condition that these coincide on the intersection of two arbitrary open sets of the covering. The sheaves which will interest us the most in this book are the constant sheaves, whose sections on U are locally constant maps with values in a fixed group G, and the sheaves of (continuous, differentiable, holomorphic) sections of a (topological, differentiable, holomorphic) vector bundle over a topological space, a differentiable manifold or a complex manifold. Let A denote the sheaf of continuous, differentiable or holomorphic functions over X, and let the term “of class A” mean continuous, differentiable or holomorphic according to A.