"Gromov–Witten theory of a space X deals with the count of the number of maps to X from a Riemann surface of a given genus which have fixed degree and whose image meets a given collection of cycles in X. They play an important role in Geometry and Physics: they carry enumerative information on X based on curve counting, they furnish a sophisticated set of invariants of the symplectic structure of X, and they make an ubiquitous appearance in manyimportant quantities of supersymmetric gauge and string theories, from effective Lagrangians to black hole microstate countings.This proposal is centered on the connection of Gromov-Witten invariants with classical integrable hierarchies of nonlinear PDEs. After the Witten-Kontsevich theorem, this has been a central, but yet poorly understood aspect of the subject. We propose to fill this gap concretely in a broad number of cases, which are at the same time of considerable physical relevance, by applying a variety of modern tools from the theory of Integrable Systems. Our aim is to provide new constructions of Integrable Systems related to Gromov-Witten theory, and apply this knowledge to address core questions in the subject, from the computation ofhigher genus invariants (corresponding to higher order quantum effects in String Theory) to the Virasoro conjecture. Special focus will be given to the applications in Physics. In doing so we will draw on and make rigorous recent insights from topological string theory."