This article is a review of recent developments in the phenomenological description of unconventional superconductivity. Starting with the BCS theory of superconductivity with anisotropic Cooper pairing, the authors explain the group-theoretical derivation of the generalized Ginzburg-Landau theory for unconventional superconductivity. This is used to classify the possible superconducting states in a system with given crystal symmetry, including strong-coupling effects and spin-orbit interaction. On the basis of the BCS theory the unusual low-temperature properties and the (resonant) impurity scattering effects are discussed for superconductors with anisotropic pairing. Using the Ginzburg-Landau theory, the authors study several bulk properties of such superconductors: spontaneous lattice distortion, upper critical magnetic field, splitting of a phase transition due to uniaxial stress. Two possible mechanisms for ultrasound absorption are discussed: collective modes and damping by domain-wall motion. The boundary conditions for the Ginzburg-Landau theory are derived from a correlation function formulation and by group-theoretical methods. They are applied to a study of the Josephson and proximity effects if unconventional superconductors are involved there. The magnetic properties of superconductors that break time-reversal symmetry are analyzed. Examples of current and magnetic-field distributions close to inhomogeneities of the superconducting order parameter are given and their physical origin is discussed. Vortices in a superconductor with a multicomponent order parameter can exhibit various topological structures. As examples the authors show fractional vortices on domain walls and nonaxial vortices in the bulk. Furthermore, the problem of the possible coexistence of a superconducting and a magnetically ordered phase in an unconventional superconductor is analyzed. The combination of two order parameters that are almost degenerate in their critical temperature is considered with respect to the phase-transition behavior and effects on the lower and upper critical fields. Because heavy-fermion superconductors—which are possible realizations of unconventional superconductivity—have been the main motivation for the phenomenological studies presented here, the authors compare the theoretical results with the experimental facts and data. In particular, they emphasize the intriguing features of the compound U${\mathrm{Pt}}_3$ and consider in detail the alloy ${\mathrm{U}}_{1-x}{\mathrm{Th}}_x{\mathrm{Be}}_{13}$.

DOI:https://doi.org/10.1103/RevModPhys.63.239