<b>Abstract</b>— <p>Two fundamental questions of modern theoretical physics—the paradox of radiant power and the issue of localized currents—are discussed in the paper. The radiant power paradox is the dilemma of whether electrodynamic systems radiate power outward or not. Mathematical analysis of the microscopic Maxwell equations describing excitation and propagation of electromagnetic fields shows that the total radiant power should be zero, whereas many classical problems considering localized currents argue the opposite. The paradox is resolved by proving limited applicability of localized-current models to calculations of energy and momentum characteristics of electromagnetic fields. The nonzero value of the total radiant power for these currents turns out to be the localized current approximation error, while the true radiant power value is always zero for physically consistent excitation currents. On the basis of analysis of physically consistent currents, an improved model of localized currents is proposed that enables physically consistent description of electromagnetic fields from the viewpoint of their energy and momentum characteristics but at the expense of causality violation in the external current-free region.</p>