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Stochastic volatility is commonly used in financial modelling to describe alternating periods of high and low fluctuations in logarithmic returns. The concept of volatility clusters can also be extended to space and space-time, and are associated with for example, variable yields across fields and volatile wind speeds. In this talk, we look at two classes of stochastic processes: a spatio-temporal Ornstein-Uhlenbeck (OU) process and a volatility modulated moving average (VMMA). Both can be seen as subclasses of a more general framework: ambit fields. While the spatio-temporal OU process is potentially a model for spatio-temporal volatility, the VMMA is a spatial model with volatility. The latter is constructed by introducing a stochastic volatility process into the Gaussian moving average (or process convolution) which is commonly used in Geostatistics. For each of these processes, we touch on theory, simulation as well as estimation methods.