We show that the model wave functions used to describe the fractional quantum Hall effect have exact representations as matrix product states (MPS).  Due to the deep relationship between these wavefunctions and the conformal field theory describing their edge, the MPSs take on a simple analytic form.  These MPSs can be implemented numerically in the orbital basis of both finite and infinite cylinders, which provide an efficient way of calculating arbitrary observables.  We extend this approach to include charged excitations and numerically compute their Berry phases. Finally, we outline how the density matrix renormalization group (DMRG) can be used to obtain non-model wavefunctions for a general microscopic Hamiltonian, and the ways to identify a topological phase from just its ground state wavefunctions.