A general non-perturbative approach to functional integrals describing systems possessing both thermal and quantum fluctuations is presented. The partition function is evaluated in the semiclassical limit by enlarging the class of paths in the functional integral to include complex-valued solutions of the classical equations of motion. Semiclassical energy eigenvalues, including tunnelling effects associated with decay of false vacua, can be obtained in this way for all states, not just the ground state. In this approach the imaginary part of an energy eigenvalue of an unstable state is directly related to physical boundary conditions at infinity; these results are compared to earlier attempts (fate of the false vacuum) to compute decay rates of metastable systems, involving an analytic continuation prescription for a negative mode of the second-order fluctuation operator. Use of complex-valued solutions obviates the need to employ the dilute-gas approximation which is generally invalid at finite temperature. The new method is checked by one-dimensional examples, and then extended to field theory, where a general expression for the partition function of a field theory is given in terms of stability angles about a complex classical solution.