During the past decade, the interaction between geometry and physics has led to tremendous advances in both fields. In particular, string theory directly led to the creation of Gromov-Witten theory, in turn yielding the solution to problems in enumerative geometry which seemed inaccessible only a few years earlier. For Calabi-Yau threefolds, Gopakumar and Vafa have defined new invariants related to the Gromov-Witten invariants using M-theory, branes, and duality in physics. These invariants, unlike the Gromov-Witten invariants, are integer-valued, and should be more directly related to our intuitive idea of "the number of holomorphic curves". In this colloquium talk, the above notions will be discussed, including a status report on the program of finding a precise mathematical formulation of these new ideas from physics.