Space weathering processes that operate in the lunar regolith modify the surfaces of lunar soil grains. Transmission electron microscope analysis of the lunar soil grains from the fine size fraction of several lunar soils show that most grains are surrounded by thin (60-200 nm thick) rims. The microstructure and chemical compositions of the rims can be used to classify rims into four broad categories: amorphous, inclusion-rich, multiple, and vesicular. Amorphous rims are noncrystalline, generally lack crystalline inclusions, show evidence for preferential sputtering of cations, and are produced largely by solar-wind irradiation damage. Inclusion-rich rims contain abundant nanometer-sized grains of Fe metal as randomly dispersed inclusions or as distinct layers embedded in an amorphous silica-rich matrix. Inclusion-rich rims are compositionally distinct from their host grains and typically contain accumulations of elements that are not indigenous to the host. Inclusion-rich rims are formed largely by the deposition of impact-generated vapors with a contribution from the deposition of sputtered ions. A continuum in the chemical and microstructural properties exists between typical amorphous rims and typical inclusion-rich rims. Multiple-rims consist of a distinct radiation-damaged layer up to 50 nm thick, that is overlain by vapor-deposited material of comparable thickness. Vesicular rims are compositionally similar to their hosts and are characterized by an abundance of small (<50 nm in diameter) vesicles concentrated in the outer 100 nm of the rims. The formation of vesicular rims is apparently due to the evolution of solar-wind implanted gases in response to a pulse-heating event. The formation of rims on lunar soils is complex and involves several processes whose effects may be superimposed. From this study, it is shown that one process does not dominate and that the relative importance of vapor-deposition is comparable to radiation-damage in the formation of rims on lunar silicate grains. The presence of rims on lunar soil grains, particularly those with nanometer-sized Fe metal inclusions, may have a major influence on the optical and magnetic properties of lunar soils.