The discussion below is for suitably small categories and objects. The aim of this paper is to present a homology theory for the category of cosheaves \(Cosh(X)=Sh^{op}(X,(\Pr o-Ab)^{op})=Sh^{op}(X,Ind- Ab^{op}),\) where Ab denotes the category of Abelian groups. For this purpose elementary categories (complete, cocomplete with a small generating family of projective objects) are shown to have enough injectives. Sheaf cohomology is defined in algebraic geometry [see \textit{A. Grothendieck}, Tôhoku Math. J., II. Ser. 9, 119-221 (1957; Zbl 0118.261)] using the fact that one has enough injectives. Furthermore, for an abelian category A with enough projectives, Ind-A is seen to be elementary. A tensor product \(\otimes: Ab\times \Pr o-Ab\to \Pr o-Ab\) with a canonical right adjoint defined via hom-functors is determined. Using the above results it follows that \(Ind\)-Ab\({}^{op}\) has enough projectives. Most of the usual properties of abelian sheaves hold for cosheaves. Homology and hyperhomology for cosheaves can, as a consequence, be defined. A link between the homology of cosheaves and the cohomology of sheaves is obtained. Some results for the homology of cosheaves on taut spaces are found. Using the tensor product defined earlier, a ``universal Kronecker product and universal cap-product'' are introduced. Finally, for the constant cosheaf \({\mathbb{Z}}\) on an HLC space, the cosheaf homology is seen to coincide with the classical singular homology.