An exotic fractionalized Fermi-liquid FL * theory of metallic systems, which combines resonant-valence-bond (RVB) state and the band of current carriers, is treated. An application of this theory to spin-liquid, antiferromagnetic and nearly antiferromagnetic systems is proposed with the use of various bosonic and fermionic representations, a comparison with perturbation theory in the s − d( f ) exchange model being performed. The topological aspects including formation of the small Fermi surface are treated. In the case of narrow bands (strong correlations), the ground state is considered as a direct product of RVB and dopon or Weng's fermion states. Examples of Kondo lattices, doped pyrochlores, and metallic β-Mn, YMn 2 , Y 1−x Sc x Mn 2 systems are discussed, analogies with copper-oxide systems being treated.