This paper is a review of low-energy positron interactions with atoms and molecules. Processes of interest include elastic scattering, electronic and vibrational excitation, ionization, positronium formation and annihilation. An overview is presented of the currently available theoretical and experimental techniques to study these phenomena, including the use of trap-based positron beam sources to study collision processes with improved energy resolution. State-resolved measurements of electronic and vibrational excitation cross sections and measurement of annihilation rates in atoms and molecules as a function of incident positron energy are discussed. Where data are available, comparisons are made with analogous electron scattering cross sections. Resonance phenomena, common in electron scattering, appear to be less common in positron scattering. Possible exceptions include the sharp onsets of positron-impact electronic and vibrational excitation of selected molecules. Recent energy-resolved studies of positron annihilation in hydrocarbons containing more than a few carbon atoms provide direct evidence that vibrational Feshbach resonances underpin the anomalously large annihilation rates observed for many polyatomic species. We discuss open questions regarding this process in larger molecules, as well as positron annihilation in smaller molecules where the theoretical picture is less clear.