  We review the mathematical formalism underlying the modelling of
stochasticity in biological systems. Beginning with a description of the system
in terms of its basic constituents, we derive the mesoscopic equations
governing the dynamics which generalise the more familiar macroscopic
equations. We apply this formalism to the analysis of two specific
noise-induced phenomena observed in biologically-inspired models. In the first
example, we show how the stochastic amplification of a Turing instability gives
rise to spatial and temporal patterns which may be understood within the linear
noise approximation. The second example concerns the spontaneous emergence of
cell polarity, where we make analytic progress by exploiting a separation of
time-scales.
