We investigate several possible notions of "canonical'' metrics that naturally arise in Hermitian non-Kähler geometry. In particular, we study an analogue of the Yamabe problem in the non-Kähler setting, concerning the existence of Hermitian metrics with constant scalar curvature with respect to the Chern connection. We also develop a moment map interpretation of the Chern scalar curvature in the locally conformally Kähler setting. Another tool for highlighting ``canonical structures'' is the Chern–Ricci flow. The long-time behavior of its solutions is expected to reflect the underlying complex structure, and we present some evidence of this in the case of compact complex surfaces. This talk is based on joint work with Simone Calamai, Mauricio Corrêa, Francesco Pediconi, Cristiano Spotti, Valentino Tosatti, and Oluwagbenga Joshua Windare.