In this talk I will discuss the stochastic dynamics of a number of problems in population biology and biochemistry using the formalism of master equations. When they contain a large number of constituents, the behaviour of these systems may be analysed using an expansion in the system size. To leading order the deterministic analogues of the models can be compared to the equations which are normally written down on phenomenological grounds. At next to leading order a simplified stochastic description is obtained. Attention will focus on systems for which the deterministic description fails to predict cycles, but where large cycles are found at next-to-leading order through a resonant amplification of demographic fluctuations. The generality and applicability of these results will be discussed.