For every compact Riemannian manifold, it is well known that the fundamental group is finitely generated. For complete non-compact Riemannian manifolds, the fundamental group possibly is not finitely generated. A natural question is: which complete Riemannian manifolds have finitely generated fundamental group? We will survey the progress in this question from Bieberbach, Cheeger-Gromoll, Gromov to more recent work by Kapovitch and Wilking, and my recent work will also be presented. No technical proofs in the talk, some elementary topology and Riemannian geometry knowledge is enough to understand most of the talk.

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