We discuss the development on geometric and analytic aspects of the Anomaly flow. Such flow naturally arises in the study of a system of equations for supersymmetric vacua of superstrings proposed independently by C. Hull and A. Strominger in 1980s. The system allows non-vanishing torsion and they incorporate terms which are quadratic in the curvature tensor. As such they are also particularly interesting from the point of view of both non-Kaehler geometry and the theory of nonlinear partial differential equations. While the complete solution of the system seems out of reach at the present time, we describe progress in developing a new general approach based on geometric flows. It turns out that the corresponding flow shares some features with the Ricci flow and preserves the conformally balanced condition of Hermitian metrics. We will begin with a brief introduction to the balanced metrics, and then discuss the analytic aspects of the Anomaly flow. In the last lecture, we will show that, on toric fibrations, the flow exists for all time and converges, recovering in this way the well-known solution obtained by J. Fu and S.T. Yau in 2006 by solving a delicate elliptic equation of Monge-Ampere type. This is joint work with D. Phong and S. Picard.