This paper presents new types of cutting-plane algorithms for integer-constrained optimization problems. The underlying idea of the method of enumerative cuts is to make use of an enumeration procedure in order to construct cutting planes that can be made arbitrarily deep. In recent publications, Balas and Young have established in somewhat different contexts that valid cutting planes can be generated from the intersection of (basic) rays with adequately constructed hyperspheres. This paper introduces a family of polyhedra to replace the sphere; these polyhedra are not tangential to the hypersphere and produce cutting planes with particular characteristics that are discussed. Constructive procedures are sketched for generating step-by-step cutting planes that become deeper and deeper at every step, until the deepest cut is obtained. This construction often relies on a partial enumeration scheme of the set of integer solutions.