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In this work, several novel algorithms for thenumerical simulation of homogeneous gas-phasecombustion chemistry are introduced andinvestigated. In the first instance, stochasticmethods for the numerical solution of systems ofordinary differential equations (ODEs) are developedbased on the theory of Markov pure-jump processes.By establishing a correspondence principle betweenstochastic and deterministic schemes, furtheralgorithms of both types are obtained. Inparticular, a scheme for the deterministic directsimulation of chemical reactions is devised. Allmethods under consideration have in common that theyare explicit, and hence conceptually very simple andeasy to implement. For three of the proposedalgorithms, detailed numerical studies of some oftheir properties such as efficiency and convergenceare performed for the high-temperature isobarichomogeneous gas-phase combustion of varioushydrocarbon fuels. In addition, the new algorithmsare compared in terms of performance to each otherand to some widespread conventional ODE-solverpackages for stiff systems.