Many environmental data sets have a continuous domain, in time and/or space, and complex features that may be poorly modelled with a stationary (in space and time) Gaussian process (GP). We adapt stochastic volatility modelling to this context, resulting in a stochastic heteroscedastic process (SHP), which is unconditionally stationary and non-Gaussian. Conditional on a latent GP, the SHP is a heteroscedastic GP with non-stationary (in space and time) covariance structure. The realizations from SHP are versatile and can represent spatial inhomogeneities. The unconditional correlation functions of SHP form a rich isotropic class that can allow for a smoothed nugget effect. We apply an importance sampling strategy to implement pseudo maximum likelihood parameter estimation for the SHP. To predict the process at unobserved locations, we develop a plug-in best predictor. We extend the single-realization SHP model to handle replicates across time of SHP realizations in space. Empirical results with simulated data show that SHP is nearly as efficient as a stationary GP in out-of-sample prediction when the true process is a stationary GP, and outperforms a stationary GP substantially when the true process is SHP. The SHP methodology is applied to enhanced vegetation index data and US NO3 deposition data for illustration.