This paper contemplates the use of robust optimization as a framework for addressing problems that involve endogenous uncertainty, i.e., uncertainty that is affected by the decision maker's strategy. To that end, we extend generic polyhedral uncertainty sets typically considered in robust optimization into sets that depend on the actual decisions. We present the derivation of robust counterpart models in this setting, and we discuss relevant algorithmic considerations for solving these models to guaranteed optimality. Besides capturing the functional changes in parameter correlations that may be induced by given decisions, we show how the use of our decision-dependent uncertainty sets allows us to also eradicate conservatism effects from parameters that become irrelevant in view of the optimal decisions. We quantify these benefits via a number of case studies, demonstrating our proposed framework's versatility to be utilized in the context of various applications. (c) 2018 Elsevier Ltd. All rights reserved.