Abstract: Statistical algorithms such as Monte Carlo integration are good candidates to run on graphics processing units. The heart of these algorithms is random number generation, which generally has been done on the CPU. In this paper we present GPU implementations of three random number generators.We show how to overcome limitations of GPU hardware that affect the feasibility and efficiency of employing a GPU-based RNG. We also present a data flow model for managing and updating substream state for each of the parallel substreams of random numbers. We show that GPU random number generators will greatly benefit from having more outputs from each thread. We discuss other hardware modifications that will be beneficial to the implementation of GPU-RNG, and we present performance measurements of our implementations.