Abstract: This talk focuses on the modeling of and portfolio optimization in financial markets with different rough models for the volatility. In the first part we deal with the (one-dimensional) stochastic volatility model of Heston, which is extended to fractional and rough market dynamics by applying the Riemann-Liouville fractional integral operator and the Marchaud fractional derivative to the classical Cox-Ingersoll-Ross process. Using a Markovian representation, followed by a reasonable quantization of the underlying probability measures, we show that it is possible to cast the problem into the classical stochastic control framework. We deduce a Feynman-Kac representation for these one-dimensional fractional and rough market models and solve the corresponding continuous time Merton portfolio optimization problems for power utility. In the second part we are again concerned with portfolio selection for an investor with power utility, but now in a multi-dimensional rough stochastic environment. In particular we investigate Merton’s, portfolio problem for different multivariate Volterra models. Based on the classical Wishart model, we introduce a new matrix-valued stochastic volatility model, where the volatility is driven by a Volterra-Wishart process. In contrast to the first part, the solution methods are now based on the calculus of convolutions and resolvents. The resulting optimal strategy can be expressed in terms of the solution of a multivariate Riccati-Volterra equation, extending existing results to the multivariate case, avoiding restrictions on the correlation structure.