This thesis is about a construction of derived functors which considerably generalises the original definition of Cartan and Eilenberg. The new derived functors can be constructed In the following circumstances:- Let C be a category with an initial object, I, and let M be a subcategory of C containing I. If F is a functor from M to a category of "based sets with structure" then the derived functors of F can be defined. The derived functors of F have domain C and their actions on objects of C are expressed as the homotopy groups of a simplicial set. This simplicial set is the nerve of a category so the derived functors can be regarded as the (topological) homotopy groups of the classifying space of a category. The 0-th derived functor constructed in this manner is the Kan extension of F. The derived functors satisfy an "acyclic model theorem". Taking M as the full subcategory of projectives of an abelian category the derived functors agree with those of Cartan and Eilenberg, as is shown using the theory of over categories and Γ-spaces. A previous important generalisation of derived functors are the cotriple derived functors. If the cotriple derived functors of F are defined then, for a suitable choice of M, the new derived functors agree with them. This shows that many well-known sequences of functors arise from the construction, including Eckmann-Hilton homotopy groups, Hochschild's K-relatlve Tor, the homology of groups and the homology of commutative algebras. The singular homology functors occur as the derived functors of the 0-th reduced singular homology functor and for these new derived functors the derived functors of the 0-th homotopy functor behave like the higher homotopy functors when applied to connected spaces.