The main theorem of this article is as follows: for each positive number V and positive integer n not equal to 3 or 4, there exists only a finite number of diffeomorphism classes containing n-dimensional complete negatively curved manifolds satisfying the inequalities \(-1<\sec tional\) curvature and volume\(<V\). The author also proves a finiteness theorem for complete (but not necessarily compact) Riemannian manifolds. He also gives an estimate on the number of homotopy types containing Riemannian manifolds satisfying the conditions of the main theorem described above.