Novel approaches to surface hopping (in the case in which surfaces cross in classically allowed regions) and surface "jumping" (in cases in which they never cross or they cross in classically forbidden regions) are discussed. Classically forbidden transitions necessarily involve discontinuous "jumps" in position or momentum or both (but so as to preserve energy). In general, the jumps are discontinuous changes in nuclear positions or momenta on the time scale of the electronic transition. After reviewing various approaches in one dimension, a phase-space approach is applied to multidimensional systems with large energy gaps, in which the traditional semiclassical approaches are more difficult to apply. The concept of jumps extends the spirit of surface hopping into new regimes.