Vertex embeddings of regular polytopes

The question of when one regular polytope (finite, convex) embedds in the vertices of another, of the same dimension, leads to a fascinating interplay of geometry, combinatorics, and matrix theory, with further relations to number theory and algebraic topology. This mainly expository paper is an account of this subject, its history, and the principal results together with an outline of their proofs. The relationships with other branches of mathematics are also explained.