A review is given of the ionization of organic moecules by monoenergetic positrons having energies in the range of 0.5–15 eV. Two mechanisms, unique to positrons, are described. If the kinetic energy of the positron is above the positronium formation threshold, such that electrons can be removed from the molecules to form free positronium atoms, the ionization/fragmentation behavior can be explained qualitatively by a modification of the Ore gap theory. To explain how positrons can ionize and fragment molecules when their kinetic energies are below the positronium formation threshold, it is necessary to assume that energy is transferred to the molecule by the annihilation process. Ionization cross sections for positrons having kinetic energies below the positronium formation threshold are sensitive to molecular size, structure and bond types. Continuing work involves a search for positronium compound formation and measurements of the kinetic energy distributions of ions.