Fc domains and CH3 domains are disclosed that bind the neonatal Fc (FcRn) receptor and are at least 99% monomeric. Monomeric Fc domain molecules and CH3 domain molecules are disclosed herein that include a monomeric Fc domain or a monomeric CH3 domain and an effector molecule. In some embodiments, the monomeric Fc or monomeric CH3 domain include amino acid substitutions and/or CDR insertions in the beta strands such that they specifically bind an antigen. Methods for using these monomeric Fc domains, monomeric CH3 domains, monomeric Fc domain molecules and CH3 domain molecules are also provided.