List of Regular Polyhedra

The regular polyhedra are those that have congruent and regular vertices, faces and edges. They are listed here, along with their Schläfli symbols.

Platonic
The five Platonic solids are the convex regular polyhedra.
 * Tetrahedron $$\{3, 3\}$$
 * Cube $$\{4, 3\}$$
 * Dodecahedron $$\{5, 3\}$$
 * Octahedron $$\{3, 4\}$$
 * Icosahedron $$\{3, 5\}$$

Kepler-Poinsot
The four Kepler-Poinsot polyhedra are concave, intersect themselves, and some have star polygons as faces rather than convex ones.
 * Great Stellated Dodecahedron $$\{5/2, 3\}$$
 * Great Icosahedron $$\{3, 5/2\}$$
 * Great Dodecahedron $$\{5, 5/2\}$$
 * Small Stellated Dodecahedron $$\{5/2, 5\}$$

Abstract
Abstract polyhedra are topologically equivalent to tilings of hyperbolic space. Therefore, they are not considered part of the nine main regular polyhedra, but are listed nonetheless.
 * Dodecadodecahedron $$\{5, 4\}$$
 * Medial Rhombic Triacontahedron $$\{4, 5\}$$
 * Medial Triambic Icosahedron $$\{6, 5\}$$
 * Ditrigonal Dodecadodecahedron $$\{5, 6\}$$
 * Excavated Dodecahedron $$\{6, 6\}$$

Spherical
Spherical polyhedra can only exist on the surface of a sphere, and are degenerate in normal Euclidean space. Therefore, they are not considered part of the nine main regular polyhedra.
 * Henagonal Henahedron $$\{1, 1\}$$
 * Henagonal Hosohedron $$\{2, 1\}$$
 * Henagonal Dihedron $$\{1, 2\}$$
 * Digonal Dihedron / Digonal Hosohedron $$\{2, 2\}$$
 * Trigonal Hosohedron $$\{2, 3\}$$
 * Trigonal Dihedron $$\{3, 3\}$$