The computer simulation of physical events begins with the selection of a mathematical model designed to abstract the features of the events thought to be of interest. Therein lies the most treacherous step in any computer simulation, a step that is often the source of the largest error in computer predictions. Model selections are generally made on the basis of empirical evidence, heuristic arguments, personal experience and judgment of the modeler, and when computed results deviate markedly from observations, little is known about how to improve results by changing or updating the model itself. Moreover, good models of physical phenomena are further corrupted when transformed via discretization into computational models that can be implemented on computers. This lecture addresses the issue of estimation and control of both modeling error and approximation error in large classes of problems in science and engineering. The idea is to define a so-called base model within a large class of models that while possibly intractable, is thought to possess sufficient structure to accurately capture all of the quantities of interest (the target outputs). This base model serves as a datum with respect to which other models can be compared. Estimates of error in quantities of interest, between the base model and other possible models are derived. Thus the error is actually a relative error between the base model and various surrogate models. Similar approaches can be used to estimate approximation error as well. A class of adaptive algorithms is developed, called goals-algorithms, to adaptively modify and control the models of a physical event to meet preset error tolerances. A number of applications are presented, including problems in multi-scale modeling of molecular and atomic systems, heterogeneous materials, random media, and quantum molecular dynamics systems.