Exponential sums over finite fields are essential ingredients in the solution of many arithmetic problems. Their study often relies on algebraic geometry, and especially on Deligne's Riemann Hypothesis over finite fields, which reveals deep structural features of these sums. In turn, one can exploit these to construct some remarkable combinatorial objects. The talk will provide a survey of these various aspects. (Joint works with A. Forey, J. Fresán and Y. Wigderson)