In this paper we provide an extensive classification of one- and two-dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black-Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions. These solutions are given by a sum of hypergeometric functions, generalizing the results obtained in a paper by Albanese et al. (Albanese, C., Campolieti, G., Carr, P. and Lipton, A., Black-Scholes goes hypergeometric. Risk Mag., 2001, 14, 99-103). For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3 / 2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.