This research monograph deals with fast stochastic simulation based on im­ portance sampling (IS) principles and some of its applications. It is in large part devoted to an adaptive form of IS that has proved to be effective in appli­ cations that involve the estimation of probabilities of rare events. Rare events are often encountered in scientific and engineering processes. Their charac­ terization is especially important as their occurrence can have catastrophic consequences of varying proportions. Examples range from fracture due to material fatigue in engineering structures to exceedance of dangerous levels during river water floods to false target declarations in radar systems. Fast simulation using IS is essentially a forced Monte Carlo procedure designed to hasten the occurrence of rare events. Development of this simu­ lation method of analysis of scientific phenomena is usually attributed to the mathematician von Neumann, and others. Since its inception, MC simula­ tion has found a wide range of employment, from statistical thermodynamics in disordered systems to the analysis and design of engineering structures characterized by high complexity. Indeed, whenever an engineering problem is analytically intractable (which is often the case) and a solution by nu­ merical techniques prohibitively expensive computationally, a last resort to determine the input-output characteristics of, or states within, a system is to carry out a simulation.