This book constructs input finite dimensional (FD) models that are amendable for numerical calculations and provides accurate representations for responses of dynamical systems to these inputs, i.e., numerical solutions of stochastic equations. It establishes conditions under which numerical solutions of these equations deliver accurate estimates of extreme responses of dynamical systems that are needed to, for example, predict extreme weather events and design reliable aircrafts. It is intended to serve a broad audience including graduate students, researchers, engineers, scientists and applied mathematicians interested in the formulation and solutions of complex stochastic problems. Includes algorithms for constructing FD models for a broad range of inputs and generating realizations of these models; Uses data-based estimates of extremes of outputs of stochastic equations based on high/low fidelity numerical solutions; Gives practical criteria for the convergence of numerical estimates of extreme system responses.