Abstract: We will give a short introduction to stochastic control in continuous time with some applications to optimal investment problems. In particular, after giving some examples of control problem which are relevant in finance and insurance, we will turn to the dynamic programming principle (DPP), which is the main tool to obtain the Hamilton-Jacobi-Bellman (HJB) partial differential equation describing the local behaviour of the value function. Under some regularity conditions, solving this HJB pde gives a method to find (at least theoretically) the optimal solution. We will apply this approach to solve some problems of optimal investment, e.g. the classic Merton problem of optimal investment and consumption.