In this work, several novel algorithms for the numerical simulation of homogeneous gas-phase combustion chemistry are introduced and investigated. In the first instance, stochastic methods for the numerical solution of systems of ordinary differential equations (ODEs) are developed based on the theory of Markov pure-jump processes. By establishing a correspondence principle between stochastic and deterministic schemes, further algorithms of both types are obtained. In particular, a scheme for the deterministic direct simulation of chemical reactions is devised. All methods under consideration have in common that they are explicit, and hence conceptually very simple and easy to implement. For three of the proposed algorithms, detailed numerical studies of some of their properties such as efficiency and convergence are performed for the high-temperature isobaric homogeneous gas-phase combustion of various hydrocarbon fuels. In addition, the new algorithms are compared in terms of performance to each other and to some widespread conventional ODE-solver packages for stiff systems.