This talk beings with a light introduction, including some historical anecdotes  to motivate the development of this Floer theoretic machinery for contact manifolds some 25 years ago.  I will discuss joint work with Hutchings which constructs nonequivariant and a family Floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over Z and capture interesting torsion information.  I will explain the need for an obstruction bundle gluing correction term in the expression of the differential in the presence of contractible Reeb orbits, which is essential even in the simple example of an ellipsoid.  I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our constructions.