Jose Voloch University of Texas at Austin Mathematics In n-dimensional projective space an arc is a set with at most n points on any hyperplane and a cap is a set with at most two points on any line. These notions coincide when n=2. We will give a survey of results on the size and characterization of the largest arcs and caps on projective spaces over finite fields of odd characteristic. An old technique of Segre (and subsequent sharpenings) relate these results to the question of estimating the size of the largest arc in the plane not contained in a conic and we will describe some results on this problem as well.