Polyhedra are basic three-dimensional objects, appearing at many levels in the algorithms of computational geometry, computer-aided design, computer vision, and robotics. What data can we use to describe a polyhedron in 3-space? What independent choices can we make when constructing a polyhedron? These are two aspects of a single question investigated in this article. The answers depend both on the level of geometry we are using (Euclidean, similarity, projective, combinatorial) and on the source of the geometric data. The constructions and representations are translated from classical and modern geometric practice. Classical theorems and techniques of Cauchy, Steinitz, Maxwell, Minkowski, and Alexandrov are transferred to this setting. Other recent geometric results are also described and a number of unsolved problems are raised.