We consider some problems concerning the geometry of the Chern connection, including: metrics with constant Chern-scalar curvature; the generalizations of the Kähler-Einstein condition to the non-Kähler setting; the convergence of the normalized Chern-Ricci flow on compact complex surfaces. The talk is based on collaborations and discussions with Simone Calamai, Francesco Pediconi, Cristiano Spotti, Valentino Tosatti.