I will present Milnor's landmark 1968 paper, "A Note on Curvature and Fundamental Group" where he proves that any finitely-generated subgroup of the fundamental group of a manifold with nonnegative Ricci curvature has at most polynomial growth. Also, we learn that if all sectional curvatures of a manifold are negative then the growth function of the fundamental group is at least exponential. I will also talk about Milnor's conjecture: If a manifold has nonnegative Ricci curvature, then its fundamental group is finitely generated, and survey recent progress in this area.