In the 1990's, theoretical physics gave rise to a new mathematical challenge: computing certain "virtual counts" of curves on manifolds. These counts, called the Gromov-Witten invariants of the manifold, model particle interactions in string theory. In principle one can determine the small-scale geometry of the universe by matching some numbers obtained in a physics lab to Gromov-Witten invariants computed in a math department (in practice, we are still waiting on the numbers from the lab). The challenge of computing Gromov-Witten invariants has motivated over two decades of mathematics, and there are still important open questions. I will discuss some explicit formulas for Gromov-Witten invariants that are available when the manifold is described by sufficiently "linear" data (representations of reductive groups), and the implications of these formulas for said open questions. These formulas also have many applications to both classical geometry and geometry motivated by physics.