We present constraints on decaying and annihilating dark matter (DM) in the 4 keV to 10 GeV mass range, using published results from the satellites HEAO-1, INTEGRAL, COMPTEL, EGRET, and the Fermi Gamma-ray Space Telescope. We derive analytic expressions for the gamma-ray spectra from various DM decay modes, and find lifetime constraints in the range 10^24-10^28 sec, depending on the DM mass and decay mode. We map these constraints onto the parameter space for a variety of models, including a hidden photino that is part of a kinetically mixed hidden sector, a gravitino with R-parity violating decays, a sterile neutrino, DM with a dipole moment, and a dark pion. The indirect constraints on sterile-neutrino and hidden-photino DM are found to be more powerful than other experimental or astrophysical probes in some parts of parameter space. While our focus is on decaying DM, we also present constraints on DM annihilation to electron-positron pairs. We find that if the annihilation is p-wave suppressed, the galactic diffuse constraints are, depending on the DM mass and velocity at recombination, more powerful than the constraints from the Cosmic Microwave Background. ; Comment: 35 pages, 15 figures; matches journal version