Saugata Basu Purdue University I will survey the basic results on various notions of complexity of semi-algebraic sets, and discuss quantitative bounds -- especially those which are important in the study of arrangements. In particular, I will explain bounds on the Betti numbers of semi-agebraic sets, including recent results which have a more refined dependence on the degree sequences of the defining polynomials, as well as bounds on the number of topological types amongst sets coming from a fixed semi-algebraic family.