This paper outlines astrophysical issues related to the long-term fate of the universe. The authors consider the evolution of planets, stars, stellar populations, galaxies, and the universe itself over time scales that greatly exceed the current age of the universe. Their discussion starts with new stellar evolution calculations which follow the future evolution of the low-mass (M-type) stars that dominate the stellar mass function. They derive scaling relations that describe how the range of stellar masses and lifetimes depends on forthcoming increases in metallicity. They then proceed to determine the ultimate mass distribution of stellar remnants, i.e., the neutron stars, white dwarfs, and brown dwarfs remaining at the end of stellar evolution; this aggregate of remnants defines the 'final stellar mass function.' At times exceeding ~1-10 trillion years, the supply of interstellar gas will be exhausted, yet star formation will continue at a highly attenuated level via collisions between brown dwarfs. This process tails off as the galaxy gradually depletes its stars by ejecting the majority and driving a minority toward eventual accretion onto massive black holes. As the galaxy disperses, stellar remnants provide a mechanism for converting the halo dark matter into radiative energy. Posited weakly interacting massive particles are accreted by white dwarfs, where they subsequently annihilate with each other. Thermalization of the decay products keeps the old white dwarfs much warmer than they would otherwise be. After accounting for the destruction of the galaxy, the authors consider the fate of the expelled degenerate objects (planets, white dwarfs, and neutron stars) within the explicit assumption that proton decay is a viable process. The evolution and eventual sublimation of these objects is dictated by the decay of their constituent nucleons, and this evolutionary scenario is developed in some detail. After white dwarfs and neutron stars have disappeared, galactic black holes slowly lose their mass as they emit Hawking radiation. This review finishes with an evaluation of cosmological issues that arise in connection with the long-term evolution of the universe. Special attention is devoted to the relation between future density fluctuations and the prospects for continued large-scale expansion. The authors compute the evolution of the background radiation fields of the universe. After several trillion years, the current cosmic microwave background will have redshifted into insignificance; the dominant contribution to the radiation background will arise from other sources, including stars, dark-matter annihilation, proton decay, and black holes. Finally, the authors consider the dramatic possible effects of a nonzero vacuum energy density.