– We use the microscopic weak coupling theory to predict the pairing state in superconductors of cubic, hexagonal, or tetragonal symmetry, where the order parameter is multicomponent, i.e., transforms according to either a 2-dimensional or a 3-dimensional representation of the crystal point group. We show that the superconducting phase usually breaks the time-reversal symmetry for singlet multicomponent superconductors. The superconducting order parameter for triplet superconductors in most cases turns out to be non-magnetic. While most superconductors are described by the Bardeen-Cooper-Schrieffer (BCS) weak coupling theory, which gives s-wave symmetry of superconducting state, in many other materials, such as He , UBe13, UPt3 , the high-Tc cuprates, Sr2RuO4 [3–5], and PrOs4Sb12 , the pairing state is known to be unconventional. (See Refs [7, 8] for a review). If the pairing state in an unconventional superconductor or superfluid transforms according to a multidimensional representation of the group, the order parameter is also multicomponent. For example, the p-wave order parameter in He is a 3 × 3 complex matrix, i.e., has 9 complex components. As a result, many different phases can arise in the superfluid state, such as the A-phase , which is not invariant with respect to rotations or time reversal operation, and the B phase, which is both rotationally and time-reversal invariant. Unconventional superconductors usually have a gapless excitation spectrum, which has important consequences for thermodynamics. Instead of describing the fascinating properties of unconventional superconductors, we refer the reader to some excellent books and reviews on the subject [2, 5, 7–10]. Although the details of the mechanism are essential for strong coupling Eliashberg-type calculations, the knowledge of the energy spectrum and the coupling constants for quasiparticles mediating superconductivity is not required  in the weak coupling approach, which is adopted below. A microscopic description of unconventional superconductors can be useful for analysis of experiments in new superconducting materials, where unconventional pairing mechanism is suspected. The usual suspects are materials with strong Coulomb correlations, such as the heavy fermions or high-Tc cuprates. In such materials conventional phonon mechanism gets suppressed . The microscopic analysis is somewhat less general then the Ginzburg-Landau (GL) approach of Refs [7,8] . For example, weak coupling BCS-type theory always gives the B-phase of superfluid He . The non-BCS spin fluctuation feedback effect, when the interaction itself depends upon the superconducting ground state, is required to stabilize the A phase .