This thesis addresses the portfolio allocation problem within a financial market featuring one riskless asset and a risky asset exhibiting rough Bergomi volatility. The objective is to maximize the expected utility of terminal wealth with respect to power utility. The volatility process in the model is driven by fractional Brownian motion and does not fit within the Markovian or semimartingale frameworks. To address this issue, we explore Markovian approximations for fractional processes and apply them to the rough Bergomi model, resulting in a multi-factor stochastic volatility model. This approach facilitates the development of a practical simulation scheme employing Gaussian quadrature and Cholesky decomposition, and allows us to address the portfolio optimization problem within a Markovian context. We solve the optimization problem using the Hamilton-Jacobi-Bellman equation, deriving an implicit solution for the case where volatility and stock return are driven by correlated Brownian motions, and providing an explicit solution for the case where they are uncorrelated. The validity of these results is further confirmed through a numerical study.