We review recent developments in our understanding of how impurities influence the electronic states in the bulk of superconductors. Our focus is on the quasi-localized states in the vicinity of impurity sites in conventional and unconventional superconductors and our goal is to provide a unified framework for their description. The non-magnetic impurity resonances in unconventional superconductors are directly related to the Yu-Shiba-Rusinov states around magnetic impurities in conventional s-wave systems. We review the physics behind these states, including quantum phase transition between screened and unscreened impurity, and emphasize recent work on d-wave superconductors. The bound states are most spectacularly seen in scanning tunneling spectroscopy measurements on high-$T_c$ cuprates, which we describe in detail. We also discuss very recent progress on the states coupled to impurity sites which have their own dynamics, and impurity resonances in the presence of an order competing with superconductivity. Last part of the review is devoted to influence of local deviations of the impurity concentration from its average value on the density of states in s-wave superconductors. We review how these fluctuations affect the density of states and show that s-wave superconductors are, strictly speaking, gapless in the presence of an arbitrarily small concentration of magnetic impurities.