Let X be a scheme of finite type over a field k. The cohomological dimension of X is the smallest integer n > 0 such that H'(X, F) = 0 for all i > n, and for all quasi-coherent sheaves F on X. There are two well-known theorems about the cohomological dimension of X. Serre's theorem states that X is affine if and only if its cohomological dimension is zero. Lichtenbaum's theorem states that if X is irreducible of dimension d, then its cohomological dimension is equal to d (the largest possible) if and only if X is proper over k. The purpose of this paper is to study situations which lie in between these two extremes. In particular, we would like to find the cohomogical dimension of projective n-space minus a closed subvariety. Our main results are the following. 1. A local vanishing theorem (3.1). This says that if A is a local ring of dimension n, and J is an ideal, such that the variety of J, V(J), meets every formal branch of Spec A in a subset of dimension > 1, then Hj3(M) = 0 for all A-modules M. This is a local analogue of Lichtenbaum's theorem, and as a corollary, it gives a new proof of Lichtenbaum's theorem. It also gives a new proof of a theorem of Nagata, which says that a normal affine surface, minus a closed subset of pure codimension one, is again affine. 2. A theorem on meromorphic functions (6.8). Let X be a closed subset of a non-singular proper scheme Z over k. Assume that X is a local complete