We study modelling issues and optimal control problems, mainly in the stochastic setting, related to advertising for new product introduction. We consider some stochastic extensions of a classical model of M. Nerlove and K. Arrow, on which we formulate and solve the mixed problem of maximizing product image (goodwill) at a given time and minimizing cumulative advertising costs, and the related problem of reaching a target level of awareness of the advertised product by a given deadline. We also allow, in some cases, budget constraints, partial observation, and discretionary launching. Then we propose new deterministic models of goodwill evolution with a spatial component, and corresponding space-time versions of the above control problems. Finally, we address optimization problems on a class of stochastic models with lags both in the state and in the control. The mathematical tools used are mainly drawn from the dynamic programming approach to optimal control, leading to the study of Hamilton-Jacobi-Bellman equations, both in finite and infinite dimensions.