Scientists and engineers use computer simulations to study relationships between a physical model's input parameters and its output predictions. However, thorough parameter studies---e.g., constructing response surfaces, optimizing, or averaging---are challenging, if not impossible, when the simulation is expensive and the model has several inputs. To enable parameter studies in these cases, the engineer may attempt to reduce the dimension of the model's input parameter space. I will (i) describe computational methods for discovering low-dimensional structures in the parameter-to-quantity-of-interest map, (ii) propose strategies for exploiting the low-dimensional structures to enable otherwise infeasible parameter studies, and (iii) review results from several science and engineering applications.