Julia Bergner 21 November 2006 Title: Understanding homotopy theories via complete Segal spaces Abstract: Given any model category, one can obtain from it a simplicial category which encodes all the homotopy-theoretic information of the original model category. Have a model structure on the category of all (small) simplicial categories is then a first step in studying the homotopy theory of homotopy theories. While this model structure does exist with appropriate weak equivalences, it is useful to consider Quillen equivalent model categories which are potentially easier to work with. There are three known alternative model categories, each with different advantages. Here we will focus on Rezk's complete Segal space model structure on the category of simplicial spaces and discuss some questions about simplicial categories which may be more easily answered in this model category.