The aim of the paper is to review quantum solid state mechanisms of nonthermal (specific) bioeffects of nonionizing radiation and to present the author's own hypothesis concerning mechanisms based on biological superconductivity. Classical and quantum mechanisms of bioeffects are compared stressing the necessity of not only considering quantum absorption, transfer, and conversion of radiation energy in biological systems, but also of appropriate systems modeling. The need is stressed for developing quantum models of the biological solid state on the supramolecular level to fill the gap between molecular and cell biology. The supramolecular models of macromolecules and enzyme complexes will be reviewed. The high-temperature superconductivity problem in organic systems will be discussed with stress on the importance of system structure and the excitation quasi-particle (phonon and electron) spectra relationship. New mechanisms of enzymatic activity assuming enzyme-substrate complex electron spectrum instability induced by electron- and phonon-mediated electron-electron interaction are proposed. Since this quantum cooperative phenomenon is the possible origin of specificity and efficiency of enzyme action it is extremely sensitive to system-generated electromagnetic fields, which gives the possibility of enzymatic regulation and also may explain some nonthermal resonant bioeffects. Local superconductivity (coherent electron states) and Josephson effects as the possible mechanisms of bioeffects are discussed.