A new method for efficiently solving global Monte Carlo particle transport problems is presented. (In these problems, flux information is desired across the entire system, not just at a small number of detector locations.) The method is based on the use of a weight window that distributes Monte Carlo particles uniformly throughout the system. This (a) ensures that all subregions of the system are adequately sampled and (b) controls the particle weights, even in subregions far from sources. The weight window is constructed from an approximate deterministic solution of the forward transport problem. It is argued that a weight window based on the forward transport solution is more appropriate for global problems than the more familiar concept of basing a weight window on an adjoint solution for source-detector problems. It is also shown that by using Monte Carlo-generated Eddington factors in deterministic solutions of the quasi-diffusion equation, one can inexpensively compute updated forward-based weight windows and obtain a more efficient global Monte Carlo calculation.