Covariant density functional theory is well established theoretical tool for the description of nuclear systems. However, there is still considerable number of important problems which have not been either addressed or resolved within its framework. In my talk, I will concentrate on one of them, namely, the physics of time-odd mean fields. Time-odd mean fields are related to nuclear magnetism in covariant density functional theory. They arise from space-like components of vector mesons and Lorentz invariance requires that their coupling strengths are identical to that of time-like components. We performed systematic investigation of these fields in non-rotating and rotating systems. The physics of these fields and physical observables strongly affected by them will be discussed in detail.