Enumerative Real Algebraic Geometry Frank Sottile 13 March 2001 DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry is concerned with real solutions to such systems, particularly a priori information on their number. Recent results in this area have, often as not, uncovered new and unexpected phenomena, and it is far from clear what to expect in general. Nevertheless, some themes are emerging. My intention in this talk is to describe the current state of knowledge, to indicate these themes, and to suggest lines of future research.