We undertake the study of an analogue of the Yamabe problem for complex manifolds. More precisely, for any conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar curvature. In this note, we set the problem and we provide an affirmative answer when the expected constant Chern scalar curvature is non-positive. In particular, this result can be applied when the Kodaira dimension
of the manifold is non-negative. Finally, we give some remarks on the positive curvature case, showing existence in some special cases and the failure, in general, of uniqueness of the solution.