A polyhedron is truncated regular if it is vertex-transitive with isosceles triangular vertex figures. Vertex transitivity means that for any two vertices of the polyhedron, there exists a translation, rotation, and/or reflection that leaves the outward appearance of the polyhedron unchanged yet moves one vertex to the other. A vertex figure is the polygon produced by connecting the midpoints of the edges meeting at the vertex in the same order that the edges appear around the vertex. There are five truncated regular polyhedra that are not self-intersecting, namely the Truncated Tetrahedron, the Truncated Octahedron, the Truncated Cube, the Truncated Icosahedron, and the Truncated Dodecahedron. When self-intersection is allowed, there are five other truncated regular polyhedra.

Johann Pitsch described all five of these polyhedra in 1881 [2]. Albert Badoureau described the Stellated Truncated Hexahedron and the Small Stellated Truncated Dodecahedron in 1881 [1].