This review examines the localization of one-dimensional nonlinear waves in an inhomogeneous multiphase medium. Particular attention is devoted to the localization of two types of waves, namely, solitary waves (domains) and switching waves that are the separation boundaries between the corresponding phases (domain walls). The localized state of such waves on both point and slowly-varying (in space) inhomogeneities is investigated. It is shown that several types of waves can become localized on inhomogeneities, and variation of external parameters may be accompanied by abrupt transitions between different types of localized waves. The stability of waves localized on inhomogeneities is examined together with various hysteresis phenomena that may occur in an inhomogeneous medium. The general results presented in the first part of the review are illustrated by examples of different physical systems, including superconductors, normal metals, semiconductors, plasmas, and chemical-reaction waves.