The formulation, validation and application of a procedure for simulating strong ground motions for use in engineering practice are described. The procedure uses empirical source functions (derived from near-source strong motion recordings of small earthquakes) to provide a realistic representation of effects such as source radiation that are difficult to model at high frequencies due to their partly stochastic behavior. Wave propagation effects are modeled using simplified Green's functions that are designed to transfer empirical source functions from their recording sites to those required for use in simulations at a specific site. The procedure has been validated against strong motion recordings of both crustal and subduction earthquakes. For the validation process we choose earthquakes whose source models (including a spatially heterogeneous distribution of the slip of the fault) are independently known and which have abundant strong motion recordings. A quantitative measurement of the fit between the simulated and recorded motion in this validation process is used to estimate the modeling and random uncertainty associated with the simulation procedure. This modeling and random uncertainty is one part of the overall uncertainty in estimates of ground motions of future earthquakes at a specific site derived using the simulation procedure. The other contribution to uncertainty is that due to uncertainty in the source parameters of future earthquakes that affect the site, which is estimated from a suite of simulations generated by varying the source parameters over their ranges of uncertainty. In this paper, we describe the validation of the simulation procedure for crustal earthquakes against strong motion recordings of the 1989 Loma Prieta, California, earthquake, and for subduction earthquakes against the 1985 Michoacán, Mexico, and Valparaiso, Chile, earthquakes. We then show examples of the application of the simulation procedure to the estimatation of the design response spectra for crustal earthquakes at a power plant site in California and for subduction earthquakes in the Seattle-Portland region. We also demonstrate the use of simulation methods for modeling the attenuation of strong ground motion, and show evidence of the effect of critical reflections from the lower crust in causing the observed flattening of the attenuation of strong ground motion from the 1988 Saguenay, Quebec, and 1989 Loma Prieta earthquakes.