We prove that in a T₂-sheaf (Y, p, B), where B is locally connected, continuation along a path is equivalent to the existence of a lift of that path. Thus we may study a general continuation theory in the context of lifts into sheaves. We show the existence of lifts of certain paths in a neighborhood of a path for which a lift exists. Using this result, we prove several homotopy lifting theorems. Finally we prove a generalization of the classical Poincaré-Volterra Theorem.