This review covers the progress in the study of vortex oscillations in rotating superfluids. The paper deals with the theory as its principal concern, but the experiments that one can compare with the theory considered are also discussed. Attention is focused mainly on the effects of crystalline order in the vortex lattice (the Tkachenko waves especially) and on the boundary problems arising in studies of vortex oscillations in finite containers. The approach is based mostly on the continuum hydrodynamic theory dealing with dense vortex arrays, and considerable attention is devoted to discussion of this theory in order to understand better the principles upon which the obtained results rest. The theory is traced from the simple description of a rotating classical fluid with continuous vorticity, through that of a perfect fluid with quantized vorticity in the form of an array of vortex lines, then the two-fluid theory of an isotropic superfluid, and finally the theory of rotating anisotropic superfluids such as ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-$A$. Applications of the theory to He II, the superfluid phases of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, and the superfluid neutron matter in pulsars are discussed.

DOI:https://doi.org/10.1103/RevModPhys.59.87