Stringy Geometry: Paleo, Meso and Neo

In this review I'll discuss how string theory has contributed and will contribute to geometry in terms of structure on the string worldsheet. The old original version, that led to mirror symmetry and Gromov-Witten invariants, consisted of Riemann surfaces mapping into a target space. The later version considered Riemann surfaces with boundaries and led to homological mirror symmetry and stability conditions on the derived category. The most recent version considers Riemann surfaces with defect lines, but the geometric applications are yet to be fully understood.