Analyzing the volatility surface is important for understanding exposure to non-linear derivative instruments. This chapter presents a first-in-literature Bayesian framework for calibrating the Heston and Rough Bergomi stochastic volatility models to the entire volatility surface using option volatility data on the FTSE/JSE All Share and the EuroStoxx 50 indices. In our approach, sparse Gaussian processes are trained offline using simulated data from the respective models to learn the pricing function, from which calibration can be performed online. We consider sparse Gaussian processes for each point on the volatility surface and multi-output (i.e., for the whole surface simultaneously) sparse Gaussian processes within our framework and compare our approach with deep neural networks. The results show that our approach produces similar accuracy to the deep neural networks, with the added benefit of showing uncertainty across the volatility surface and requiring fewer hyperparameter configurations. Furthermore, we observe that the Bayesian calibration becomes more certain as we decrease maturity and at lower relative moneyness levels for a given maturity.