This thesis deals with stochastic volatility for financial markets with a special focus on the recent paradigm of rough volatility. We start with an investigation of how the volatilityof-volatility of the S&P 500 index depends on volatility. Our main conclusion is that volatility behaves more like a log-normal model than a square-root model. Moreover, we find that an accurate specification of the level-dependence matters for the predictive quality, including for the effective hedging of options. Next, we propose the hybrid multifactor scheme for the simulation of stochastic Volterra equations with completely monotone kernels, rough volatility especially. We prove convergence and develop efficient methods for computation of the VIX index for a number of volatility models: multifactor Volterra Bergomi, quadratic Volterra Heston, generalised CEV Volterra. We observe good numerical convergence except for a specific parameter choice under rough Heston where a large positivity bias appears. We then look at the problem of calibrating to SPX
options and jointly to SPX and VIX options. The key observations are as follows: We find that the one-factor rough Bergomi model falls short on solving the SPX calibration problem in two ways: (1) it fails to sufficiently separate the volatility-of-volatility that is implied by option prices at short and long expiries, (2) it fails to create a term structure of smile (a)symmetry; the latter is needed as we find short-term option smiles to be more symmetric, generally. We propose an alternative volatility model driven by two Ornstein-Uhlenbeck processes that uses a non-standard transformation function. We demonstrate that our model can calibrate almost perfectly to SPX options and very well to SPX and VIX options jointly. This suggests that the SPX and VIX options markets can largely be reconciled with two-factor classical volatility, all without roughness and jumps.