Chemical reactions are center-stage in systems biology. Due to low copy numbers and constrained environments inside biological cells or organelles, however, biological reaction kinetics in systems biology is frequently not appropriately described by continuous models, such as differential equations. The chemical master equation (CME) provides an exact mesoscopic description, including higher-order fluctuation kinetics. Numerically sampling from the CME traditionally made use of Gillespie-type algorithms. These algorithms, however, are computationally expensive, hampering large simulations and parameter identification. We present a new class of exact stochastic simulation algorithms (SSA) to sample from the CME in linear or constant time. The algorithms are based on the concept of partial propensities, which allow factorizing the problem into less complex ones. We further show applications of this class of algorithms to studying the effects of fluctuations in mesoscopic reaction networks, as frequently found in systems biology. We demonstrate a novel concentration inversion effect in monostable reaction networks and present applications to parameter inference in network models from noisy experimental data.