This thesis explores the topic of maximum mass stability of neutron stars. The outer structure is detailed and explores nuclear pasta phases, the neutron drip line, and density transitions of matter in the crust and atmosphere layers. Other discussion points include superfluids in the crust and core, vortex roles in neutron stars, and magnetic field effects on the EOS in neutron stars. The inner core is studied in much more detail due to its significant role in EOS. The variety of stars include pion condensate stars, kaon condensate stars, npeu stars, npeu stars with the inclusion of hyperons, quark-hybrid stars, and strange stars. Included with these is a description of nucleon-nucleon, nucleon-nucleon-nucleon interactions, the appearance factors that affect hyperon species, and the formation process of kaons, pions, quarks, and hyperons. The ending EOS are compared with their maximum mass values to determine which ones are likely to limit the mass of neutron stars.