We present an efficient and rigorous numerical procedure for calculating the elastodynamic response of a fault subjected to slow tectonic loading processes of long duration within which there are episodes of rapid earthquake failure. This is done for a general class of rate- and state-dependent friction laws with positive direct velocity effect. The algorithm allows us to treat accurately, within a single computational procedure, loading intervals of thousands of years and to calculate, for each earthquake episode, initially aseismic accelerating slip prior to dynamic rupture, the rupture propagation itself, rapid post seismic deformation which follows, and also ongoing creep slippage throughout the loading period in velocity-strengthening fault regions. The methodology is presented using the two-dimensional (2-D) antiplane spectral formulation and can be readily extended to the 2-D in-plane and 3-D spectral formulations and, with certain modifications, to the space-time boundary integral formulations as well as to their discretized development using finite difference or finite element methods. The methodology can be used to address a number of important issues, such as fault operation under low overall stress, interaction of dynamic rupture propagation with pore pressure development, patterns of rupture propagation in events nucleated naturally as a part of a sequence, the earthquake nucleation process, earthquake sequences on faults with heterogeneous frictional properties and/or normal stress, and others. The procedure is illustrated for a 2-D crustal strike-slip fault model with depth-variable properties. For lower values of the state-evolution distance of the friction law, small events appear. The nucleation phases of the small and large events are very similar, suggesting that the size of an event is determined by the conditions on the fault segments the event is propagating into rather than by the nucleation process itself. We demonstrate the importance of incorporating slow tectonic loading with elastodynamics by evaluating two simplified approaches, one with the slow tectonic loading but no wave effects and the other with all dynamic effects included but much higher loading rate. ¿ 2000 American Geophysical Union