In recent years, the availability of intense positron beams allowed for an increase in the number of ionization experiments. The first experiments, which studied noble gases, were followed by measurements of the ionization cross sections for other atoms and molecules and for triple differential cross section measurements. Our work used various theoretical models to reproduce the existing experimental cross sections. For the calculation of the total ionization cross sections of atoms and molecules we explored the use of distorted-wave models, containing a realistic description of the final state of the system and including distortion and polarization effects. Recently, we expanded our work to the study of the electron-capture-in-the-continuum (ECC) phenomenon in positron impact ionization of helium and molecular hydrogen. We employed a 3C theoretical model in representing the final state of the ionization system, which allowed us to obtain good agreement with the existing "triple differential cross section" experiments.