The notion of a generalized Kahler (GK) structure was introduced in the early 2000's by Hitchin and Gualtieri in order to provide a mathematically rigorous framework of certain nonlinear sigma model theories in physics. Since then, the subject has developed rapidly. It is now realized, thanks to more recent works of Hitchin, Goto, Gualtieri, Bischoff and Zabzine, that GK structures are naturally attached to Kahler manifolds endowed with a holomorphic Poisson structure. Inspired by Calabi's program in Kahler geometry, which aims at finding a "canonical" Kahler metric in a fixed deRham class, I will present in this talk an approach towards a “generalized Kahler" version of Calabi's problem motivated by an infinite dimensional moment map formalism, and using the Bismut-Ricci flow introduced by Streets and Tian as analytical tool. As an application, we give an essentially complete resolution of the problem in the case of a toric complex Poisson variety. Based on joint works with J. Streets and Y. Ustinovskiy.