This review deals with the properties of superconductors with competing ele
ctron spectrum instabilities, namely, charge-density waves (CDWs) and spin-
density waves (SDWs). The underlying reasons of the electron spectrum insta
bility may be either Fermi surface nesting or the existence of Van Hove sad
dle points for lower dimensionalities. CDW superconductors include layered
dichalcogenides, NbSe3, and compounds with the A15 and C15 structures among
others. There is much evidence to show that high-T-c oxides may also belon
g to this group of materials. The SDW superconductors include URu2Si2 and r
elated heavy-fermion compounds, Cr-Re alloys and organic superconductors. W
e review the experimental evidence for CDW and SDW instabilities in a wide
range of different superconductors, and assess the competition between thes
e instabilities of the Fermi surface and the superconducting gap. Issues co
ncerning the superconducting order parameter symmetry are also touched upon
. The accent is put on establishing a universal framework for further theor
etical discussions and experimental investigations based on an extensive li
st of available and up-to-date references.