Neutralino dark matter is generally assumed to be relatively heavy, with a mass near the electroweak scale. This does not necessarily need to be the case, however. In the next-to-minimal supersymmetric standard model (NMSSM) and other supersymmetric models with an extended Higgs sector, a very light CP-odd Higgs boson can naturally arise making it possible for a very light neutralino to annihilate efficiently enough to avoid being overproduced in the early Universe. In this article, we explore the characteristics of a supersymmetric model needed to include a very light neutralino, 100 MeV<m{sub {chi}}{sub -tilde1{sup 0}}<20 GeV, using the NMSSM as a prototype. We discuss the most important constraints from Upsilon decays, b{yields}s{gamma}, B{sub s}{yields}{mu}{sup +}{mu}{sup -} and the magnetic moment of the muon, and find that a light bino or singlino neutralino is allowed, and can be generated with the appropriate relic density. It has previously been shown that the positive detection of dark matter claimed by the DAMA collaboration can be reconciled with other direct dark matter experiments such as CDMS II if the dark matter particle is rather light, between about 6 and 9 GeV. A singlino or binolike neutralino could easily fall within this range of massesmore » within the NMSSM. Additionally, models with sub-GeV neutralinos may be capable of generating the 511 keV gamma-ray emission observed from the galactic bulge by the INTEGRAL/SPI experiment. We also point out measurements which can be performed immediately at CLEO, BABAR, and Belle using existing data to discover or significantly constrain this scenario.« less