This chapter reviews recent developments to manage aircraft arrivals in the context of extended arrival manager systems, for which uncertainty is significant when predicting expected times to start the approach phase and landing times. An original high-level multi-stage stochastic optimization formulation, considering several air network points of interest, is first introduced, taking account of practical operational constraints. The remaining of the chapter focuses on the two-stage special case, which corresponds to recent studies on the aircraft arrival management problem. A landing order is decided at a specific air network point known as the initial approach fix, or IAF (first stage), and a recourse cost is proposed so as to ensure that aircraft separation constraints are satisfied at the landing runway (second stage). Multiple possible IAF points are considered as well as the possibility to delay the departure of on-ground aircraft. Finally, this study proposes new analyses (validation score and impact of inclusion of chance constraints in the first stage) of numerical experiments performed on realistic instances based on Paris-Charles de Gaulle arrival data. We discuss numerical results and exhibit that the stochastic solutions are more robust than their deterministic counterparts.