We give an overview of the theory and methods involved in portfolio optimizat- ion problems with partial observation. By «partial observation», we mean that the drift process and the driving Brownian motion appearing in the stochastic differential equation for the security prices are not directly observable for investors in the market. The history of security prices is assumed to constitute the only information available to investors and the investment processes are then required to be adapted to the natural filtration of the price processes. In the complete market case, we obtain the optimal portfolio rule for a Bayesian investor and when the unobservable stock drift is modelled as a Gaussian process. We also consider the case of incomplete market and characterize the optimal investment policies when price process of risky assets follows a stochastic volatility model.