Various experimental methods based on positron annihilation have evolved into important tools for researching the structure and properties of condensed matter. In particular, positron techniques are useful for the investigation of defects in solids and for the investigation of solid surfaces. Experimental methods need a comprehensive theory for a deep, quantitative understanding of the results. In the case of positron annihilation, the relevant theory includes models needed to describe the positron states as well as the different interaction processes in matter. In this review the present status of the theory of positrons in solids and on solid surfaces is given. The review consists of three main parts describing (a) the interaction processes, (b) the theory and methods for calculating positron states, and (c) selected recent results of positron studies of condensed matter.
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