This manuscript is based upon lectures given at Princeton University during the fall semester of 1971-72. The central theme is Artin's braid group, and the many ways that the notion of a braid has proved to be important in low dimensional topology.Chapter 1 is concerned with the concept of a braid as a group of motions of points in a manifold. Structural and algebraic properties of tht braid groups of two manifolds are studied, and systems of defining relations are derived for the braid groups of the plane and sphere. Chapter 2 focuses on the connections between the classical braid group and the classical knot problem. This is an area of research which has not pro-gressed rapidly, yet there seem to be many interesting questions. The basic results are reviewed, and we then go on to prove an important theorem which was announced by Markov in 1935 but never proved in detail. This is followed by a discussion of a much newer result, Garside's solution to the conjugacy problem in the braid group. The last section of Chapter 2 explores some of the possible implications of the Garside andMarkov theorems.In Chapter 3 we discuss matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. In Chapter 4, we give an overview of recent results on the connections between braid groups and mapping class groups of surfaces. Finally, in Chapter 5, we discuss briefly the theory of "plats. The Appendix contains a list of problems. All are of a research nature, many of unknown difficulty.