Rates of pycnonuclear reactions in ultradense fluids and solids are evaluated by taking account of recent progress in quantum-statistical formulations of the equations of state and phase transitions in dense matter. New theoretical results are summarized for the reaction rates and the enhancement factors, obtained through systematic studies of the screening effects by relativistic and nonrelativistic electrons, as well as of the thermodynamic properties of dense matter resulting from internuclear cohesion in electron-screened Coulombic fluids and solids; the results prove to be a significant improvement over those in a previous review [S. Ichimaru, Rev. Mod. Phys. 65, 255 (1993)]. On the basis of these theoretical developments, coupled with renovated experiments in ultrahigh-pressure metal physics, outstanding issues of nuclear reactions in stellar and planetary interiors and in terrestrial settings are explored, with inertial-confinement-fusion experiments, and for a novel scheme of fusion studies in dense liquid-metallic proton-deuteron mixtures. (C) 1999 American Institute of Physics. [S1070-664X(99)01807-8].