Stochastic differential equations provide a useful means of introducing stochasticity into models across a broad range of systems from chemistry to population biology. However, in many applications the resulting equations have so far proved intractable to direct analytical solution. Numerical approximations, such as the Euler scheme are therefore a vital tool in exploring model behaviour. Unfortunately, current results concerning the convergence of such schemes impose conditions on the drift and diffusion coefficients of the stochastic differential equation, namely the linear growth and global Lipschitz conditions, which are often not met by systems of interest. In this paper we relax these conditions and prove that numerical solutions based on the Euler scheme will converge to the true solution of a broad class of stochastic differential equations. The results are illustrated by application to a stochastic Lotka-Volterra model and a model of chemical auto-catalysis, neither of which satisfy either the linear growth nor the global Lipschitz conditions.