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In order to capture important subcellular dynamics, researchers in computational biology have begun to turn to mesoscopic models in which molecular interactions at the gene level behave as discrete stochastic events. While the trajectories of such models cannot be described with deterministic expressions, the probability distributions of these trajectories can be described by the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could only be solved analytically in the most trivial of problems, and the CME has been analyzed almost exclusively with kinetic Monte Carlo (KMC) algorithms. However, concepts from linear systems theory have enabled the finite state projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to KMC simulations. In this paper, we review the FSP approach and introduce a variety of systems-theory-based modifications and enhancements to the FSP algorithm. Notions such as observability, controllability, and minimal realizations enable large reductions and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time-scale dynamics onto a slowly varying manifold of much smaller dimension. We also present a powerful new reduction approach, in which we perform computations on a small subset of configuration grid points and then interpolate to find the distribution on the full set. The power of the FSP and its various reduction approaches is illustrated on few important models of genetic regulatory networks.