One of the famous 23 mathematics problems Hilbert formulated in 1900, the problem asks for a topological description of Lie groups without any direct reference to smooth structure. In notes from a graduate course he taught in 2011 with some extra material from his blog, Tao develops the machinery needed to solve Hilbert's fifth problem, and then uses it to classify approximate groups and finally to develop applications such as Gromov's celebrated theorem. The first part of the book can be used in a one-quarter or one-semester advanced graduate course. It assumes familiarity with basic graduate real analysis and linear algebra. The second part contains additional material that may be but need not be used for the course. Annotation ©2014 Ringgold, Inc., Portland, OR (protoview.com)