The use of surrogate models to approximate computationally expensive simulation models, e.g., large comprehensive finite element models, is widespread. Typical uses of surrogate models include design, optimization, sensitivity analysis and/or uncertainty quantification. A surrogate model is defined by a postulated functional form, and values for the surrogate model parameters are estimated using results from a limited number of solutions to the comprehensive model. In general, there may be multiple surrogate models, each defined by possibly a different functional form, consistent with the limited data from the comprehensive model. We refer to each as a candidate surrogate model. Methods are developed and applied to select the optimal surrogate model from the collection of candidate surrogate models. One approach is to select the surrogate model that best fits the data provided by the comprehensive model, regardless of its intended use. The proposed approach applies techniques from decision theory, where postulated utility functions are used to account for the model use within the selection process. Three applications are presented to illustrate the methods. These include surrogate model selection for the purpose of: (1) estimating the minimum of a deterministic function, (2) the design under uncertainty of a simple oscillator, and (3) the uncertainty quantification of a complex engineering system subject to a severe shock and vibration environment.