Wind energy has so far been the renewable energy with the most successful development throughout Europe, while the most new capacities installed over the last year have been in the US. The stochastic nature of wind power generation uncovers new challenges for economic and safe management of power systems and electricity markets. Forecasting of wind power generation is recognized as making a significant contribution in that sense. Until a few years ago focus was almost uniquely given to the development and improvement of point forecasting methods, thus assuming a stationary and unconditional quadratic loss function of the forecast users. Recently however, a number of research groups has concentrated efforts on developing a probabilistic view of wind power forecasting, based either on physical concepts or on purely statistical methods, or finally on a combination of both. Decision-theoretic aspects related to the use of wind power forecasts will be considered for motivating such developments. In parallel, the practical motivations for probabilistic forecasting at various spatial (i.e., local or regional) and temporal (i.e., from few minutes to few days ahead) scales will be detailed, as well as a thorough characterization of wind power forecast uncertainty. A description of a number of nonparametric density forecasting methods developed over the last few years will be given, with particular emphasis on quantile regression based methods and on methods using meteorological ensemble forecasts as input. The quality (in a forecast evaluation sense) of such probabilistic forecasting methods will be discussed, as well as their value, i.e. the benefits resulting from their use. Finally, a discussion on operational aspects will attempt at explaining why probabilistic forecasts may not become a customary product in the short-term, even though significant benefits from their use have been demonstrated for a number of real-world decision-making problems.