The processes undergone by positrons in the interstellar medium (ISM) from the moments of their birth to their annihilation are examined. Both the physics of the positron interactions with gases and solids (dust grains), and the physical conditions and characteristics of the environments where the processes of energy loss, positronium formation, and annihilation taking place, are reviewed. An explanation is given as to how all the relevant physical information are taken into account in order to calculate annihilation rates and spectra of the 511 keV emission for the various phases of the ISM; special attention is paid to positron interactions with dust and with polycyclic aromatic hydrocarbons. An attempt is made to show to what extent the interactions between positrons and interstellar dust grains are similar to laboratory experiments in which beams of slow positrons impinge upon solids and surfaces. Sample results are shown for the effect of dust grains on positron annihilation spectra in some phases of the ISM which, together with high resolution spectra measured by satellites, can be used to infer useful knowledge about the environment where the annihilation is predominantly taking place and ultimately about the birth place and history of positrons in the Galaxy. The important complementarity between work done by the astrophysical and the solid-state positron communities is strongly emphasized and specific experimental work is suggested which could assist the modeling of the interaction and annihilation of positrons in the ISM. (c) 2005 Elsevier B.V. All rights reserved.