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Historically, most quantitative seismological analyses have been based on the assumption that earthquakes are caused by shear faulting, for which the equivalent force system in an isotropic medium is a pair of force couples with no net torque (a 'double couple,' or DC). Observations of increasing quality and coverage, however, now resolve departures from the DC model for many earthquakes and find some earthquakes, especially in volcanic and geothermal areas, that have strongly non-DC mechanisms. Understanding non-DC earthquakes is important both for studying the process of faulting in detail and for identifying nonshear-faulting processes that apparently occur in some earthquakes. This paper summarizes the theory of 'moment tensor' expansions of equivalent-force systems and analyzes many possible physical non-DC earthquake processes. Contrary to long-standing assumption, sources within the Earth can sometimes have net force and torque components, described by first-rank and asymmetric second-rank moment tensors, which must be included in analyses of landslides and some volcanic phenomena. Non-DC processes that lead to conventional (symmetric second-rank) moment tensors include geometrically complex shear faulting, tensile faulting, shear faulting in an anisotropic medium, shear faulting in a heterogeneous region (e.g., near an interface), and polymorphic phase transformations. Undoubtedly, many non-DC earthquake processes remain to be discovered. Progress will be facilitated by experimental studies that use wave amplitudes, amplitude ratios, and complete waveforms in addition to wave polarities and thus avoid arbitrary assumptions such as the absence of volume changes or the temporal similarity of different moment tensor components.