Continental and submarine hydrothermal systems are commonly found around the world. Similar systems that sustain water or other fluids are also likely to exist in planetary bodies throughout the solar system. Also, terrestrial submarine systems have been suggested as the locations of the first life on Earth and may, therefore, provide indications of where to find life on other planetary bodies. The study of these systems is vital to the understanding of planetary heat transfer, chemical cycling, and biological processes; hence hydrothermal processes play a fundamental role in planetary evolution. In this thesis, three particular types of hydrothermal systems are investigated through the development of mathematical models: (1) terrestrial low-temperature diffuse flows at mid-oceanic ridges (MORs), (2) submarine near-axis convection on Earth, and (3) convection driven by magmatic intrusives on Mars. Model set-ups for all systems include a two-dimensional space with a vertical, hot wall, maintained at constant temperature, located adjacent to a water-saturated porous medium at a lower temperature. By assuming that convection occurs vigorously and within a thin layer next to the hot wall, boundary layer theory is applicable. The models provide steady-state, single-phase estimates of the total heat and mass transfer rates in each scenario over permeability ranges of 10^-14 m² to 10^-10 m² for the submarine systems and 10^-14 m² to 10^-8 m² for the Martian systems. Heat output results derived from the boundary layer model suggest that diffuse flow on MORs contributes 50% or less of heat output to the ridge system, which falls at the low end of observations. For the near-axis model, results found that heat transfer in the hydrothermal boundary layer was greater than the input from steady state generation of the oceanic crust by seafloor spreading. This suggests that the size of the mushy zone evolves with time. Heat output and fluid flux calculations for Martian systems show that fluid outflow adjacent to a single intrusion is too small to generate observed Martian surface features in a reasonable length of time.