During the last decades positron annihilation has become a very powerful tool for the investigation of polymers. In particular, positron annihilation lifetime spectroscopy (PALS) yields valuable information on free volume and other properties. The present invited paper gives examples from recent research of the Kiel group. Generally, the so-called standard model, developed by Tao and Eldrup, is used to determine the size of free volume holes from the ortho-positronium (o-Ps) lifetime tau(oPs). Despite its success, the model resorts to several assumptions, including a spherical hole shape. Although the deviations from spherical shape are significant for holes above the size of positronium, average hole sizes V(h), determined by the standard model from T,p, show a good correlation with diffusivities D of inert gas molecules when plotted as 1/V(h) vs logD, as predicted by the free volume approach. The correlation can further be improved by taking into account the cohesive energy density of the polymers. The o-Ps intensity I(o-Ps) is often taken as a measure of the hole density. However, I(o-Ps) is also affected by the Ps formation probability and drops during mechanical milling of polymers due to formation of free radicals by chain scission, for instance. I(o-Ps), is also seen to change during phase separation in polymer blends. This can be explored to detect both, the binodal and the spinodal decomposition, already at the initial stage which is not easily accessible by other techniques. PALS was also used to study thermosets. Here we show in-situ results on the cross-linking of an epoxy resin. Finally, we demonstrate the benefits of the positron beam technique which allows investigations of polymer thin films and surfaces. For example, very recent results, obtained at the positron beam in Munich, on the structure and dynamics of epoxy films as function of film thickness will be presented.