Odds of a dimpled die, out of 10,000 tosses, using a Markov Chain model. This page contains a complete list of all possible Fair Dice.  First off, objects which are ISOHEDRAL are fair.  To be isohedral, each face must have the same relationship with all other faces, and each face must have the same relationship with the center of gravity.  Thus, each object has a Group structure (a Group Theory topic).  Using Group Theory and Euler's formula (Vertices + Faces - Edges = 2), it can be proven that the following list contains all possible isohedra.  An interactive version of this page is at the CRC Concise Encyclopedia of Mathematics.  A good explanation of the math behind all this can be found at Klaus Æ. Mogensen's website.

It is important to fully understand the details behind the games we play and the odds and probabilities. Now that you can play dice games online, understanding the odds becomes a software issue. There is a big difference between playing online roulette and roulette at a table. Either way by studying the game you will always get the best results.

Regular Tetrahedron	Isosceles Tetrahedron	Scalene Tetrahedron
Cube	Octahedron	Regular Dodecahedron
Octahedral Pentagonal Dodecahedron	Tetragonal Pentagonal Dodecahedron	Rhombic Dodecahedron
Trapezoidal Dodecahedron	Triakis Tetrahedron	Regular Icosahedron
Hexakis Tetrahedron	Tetrakis Hexahedron	Triakis Octahedron
Trapezoidal Icositetrahedron	Pentagonal Icositetrahedron	Dyakis Dodecahedron
Rhombic Triacontahedron	Hexakis Octahedron	Triakis Icosahedron
Pentakis Dodecahedron	Trapezoidal Hexecontahedron	Pentagonal Hexecontahedron
Hexakis Icosahedron 120 sides	Triangular Dihedron Move points up/down - 4N sides	Basic Triangular Dihedron 2N sides
Trigonal Trapezohedron Asymmetrical sides -- 2N Sides	Basic Trigonal Trapezohedron Sides have symmetry -- 2N Sides	Triangular Dihedron Move points in/out - 4N sides

Note to gaming companies:  I'd love to see someone make the exotic dice in the list above. It would be the best gambling game since Texas Hold'em poker. Imagine the odds.

Can a non-isohedral fair die exist?  Consider a pyramid made from 4 isosceles triangles and a square.  If the pyramid is short and fat, the square face will be landed upon more than a fifth of the time.  If the pyramid is tall and thin the square face will be landed upon less than a fifth of the time.  Is there a height where the square face will be landed upon exactly one fifth of the time???
Yes, for a given set of conditions.  If you knew the height, force, elasticity, and throwing method, you could find the right height.  However, once the conditions changed, the die would no longer be fair.  (NOTE:  I have a strong argument for this, but no proof.)

If you find the polyhedrons above interesting, you're bound to enjoy the Pavilion of Polyhedreality by George Hart.

There are 6 regular 4-dimensional object.  Peek is a fascinating and very pretty piece of software that allows looking at 3-D cross-sections of complex 4-D objects.  The pictures are beautiful!