We describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up to 12 vertices and up to 26 edges, simplical polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, and bipartite polyhedra with up to 22 vertices.

The results of the enumeration were used to systematically search for certain minimal non-Hamiltonian polyhedra. In particular, the smallest polyhedra satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given.