Given a set of objects S, a spatio-temporal window query q retrieves the objects of S that will intersect the window during the (future) interval q_T. A nearest neighbor query q retrieves the objects of S closest to q during q_T. Given a threshold d, a spatio-temporal join retrieves the pairs of objects from two datasets that will come within distance d from each other during q_T. In this article, we present probabilistic cost models that estimate the selectivity of spatio-temporal window queries and joins, and the expected distance between a query and its nearest neighbor(s). Our models capture any query/object mobility combination (moving queries, moving objects or both) and any data type (points and rectangles) in arbitrary dimensionality. In addition, we develop specialized spatio-temporal histograms, which take into account both location and velocity information, and can be incrementally maintained. Extensive performance evaluation verifies that the proposed techniques produce highly accurate estimation on both uniform and non-uniform data.