The physics of the processes at the edge of magnetic fusion devices is multifaceted and exhibits complex, nonlinear synergistic effects. Even though this region occupies only a small portion of the whole device, it plays a crucial role in overall plasma confinement, heat exhaust, and plasma–wall interactions. The latter affects not only the performance but also the lifetime of plasma-facing components and therefore remains an outstanding challenge for future fusion reactors. At the edge of fusion devices, researchers are dealing with phenomena including classical and anomalous plasma transport, atomic physics effects, and physics of plasma-facing material under strong irradiation by particle and energy fluxes. The diversity of edge physics makes it particularly attractive for young scientists. Working in this field, they can find endless possibilities to demonstrate their talents and creativity. This short review describes just some of the basic scrape-off layer and divertor plasma phenomena including divertor plasma detachment, intermittent bursts of anomalous cross-field plasma transport, plasma–material interactions, and dust in fusion plasmas, which are of particular interest in relation to fusion reactors.