We review the basic features of oscillations observed at different height levels in the sunspot atmosphere, moreover, various possibilities for a theoretical interpretation are discussed. In the umbra oscillation power is concentrated in severa] period bands
(3 min., 5 min., and ≥ 20 min.) which on their part are composed of closely packed peaks. The observed amplitudes and phases of velocity and of intensity oscillations depend in a characteristic way on the period and on the height. These features are used to look for the most probable physical mechanisms which could produce the different modes: At subphotospheric depths two independent resonators are acting. A resonator for slow, quasi-transveree waves can explain the lifetimes of umbral dots (≥ 20 min.), while a resonator for fast (acoustic), quasi-longitudinal waves could result in the umbral 5-min. oscillations. The acoustic resonator strongly couples with the slow-mode longitudinal resonator at photospheric and chromospheric heights, the latter produces the resonance peaks in the 3-min. band. Running penumbral waves can be explained by the transformation of 5-min, waves from the convective zone in the almost horizontal magnetic field. The interpretation of oscillations provides a new method of probing not only subphotospheric, but also atmospheric layers of sunspots (e.g., of determining temperature gradients), thus completing customary spectroscopic diagnostics.