Condensed matter physicists have long sought a realistic two dimensional (2D) magnetic system whose ground state is a spin liquid - a zero temperature state in which quantum fluctuations have melted away any form of magnetic order. The nearest-neighbor S = 1/2 Heisenberg model on the kagome lattice has seemed an ideal candidate, but in recent years some approximate numerical approaches to it have yielded instead a valence bond crystal. We have used the density matrix renormalization group (DMRG) to perform very accurate simulations on numerous cylinders with circumferences up to 12 lattice spacings, total sizes up to 400-500 sites, finding instead of the valence bond crystal a singlet-gapped spin liquid with substantially lower energy that appears to have Z2 topological order. In my thesis, we will focus on the techniques to apply DMRG efficiently to study 2D frustrated quantum systems, and our results, through a combination of very low energy, short correlation lengths and corresponding small finite size effects, and consistent behavior on many cylinders, provide strong evidence that the 2D ground state of this model is a gapped spin liquid. The spin liquid can be viewed as a melted valence bond crystal formed from 8 site diamond loops and dimers with a 12 site unit cell, called the diamond pattern. We will show that the diamond pattern appears naturally when enhancing the bonds on the resonant loops. Also we will study one of the narrowest cylinders, with a circumference of 4 lattice spacings which accommodates the diamond pattern, but for which the spin liquid ground state, while metastable state in DMRG, is higher in energy than another state with ``a topological string". The peculiar behavior of this narrow cylinder is presumably due to short resonance loops around the cylinder.