Quantum-mechanical PT-symmetric theories associated with complex cubic potentials such as V = x(2) + y(2) + igxy(2) and V = x(2) + y(2) + z(2) + igxyz, where g is a real parameter, are investigated. These theories appear to possess real, positive spectra. Low-lying energy levels are calculated to very high order in perturbation theory. The large-order behavior of the perturbation coefficients is determined using multidimensional WKB tunneling techniques. This approach is also applied to the complex Henon-Heiles potential V = x(2) + y(2) + ig(xy(2) - (1/3)x(3)). (C) 2001 Elsevier Science B.V. All rights reserved.