Summary form only given. In this talk we summarize some of our work in creating computational infrastructure to enable Monte Carlo computations in serial, parallel, distributed, and grid-based environments. We begin with a brief overview of the scalable parallel random number generators (SPRNG) library. This provides high quality pseudorandom numbers in all the above environments. We then discuss specific grid services for Monte Carlo that we recently developed. These services reduce wall clock time and improve the trustworthiness and integrity of grid-based computations. We then discuss quasirandom numbers based on scrambling in this context. Finally, we present results that differ for quasi-Monte Carlo methods on the grid from those presented for pseudorandom numbers.