The problem of modeling tunneling phenomena in more than one dimension is examined. It is found that existing techniques are inadequate in a wide class of situations, due to their inability to deal with concurrent classical motion. The generalization of these methods to allow for complex momenta is shown, and improved techniques are demonstrated with a selection of illustrative examples. Possible applications are presented. Bowcock, Peter and Gregory, Ruth Unspecified Center DE-FG02-90ER-40560; NSF PHY-90-00036; NAGW-1340...