Saugata Basu Purdue University I will survey the known results on bounding the topological complexities of semi-algebraic as well as more general definable sets. Starting from the first results in this direction due to Oleinik and Petrovskii, some of these results have been used in several different ways in discrete and computational geometry as well as in other areas of mathematics an computer science. I will state what is known, give hints on how the results are proved, and mention some open questions.