  Quantum criticality due to the valence transition in some Yb-based heavy
fermion metals has gradually turned out to play a crucial role to understand
the non-Fermi liquid properties that cannot be understood from the conventional
quantum criticality theory due to magnetic transitions. Namely, critical
exponents giving the temperature (T) dependence of the resistivity \rho(T), the
Sommerfeld coefficient, C(T)/T, the magnetic susceptibility, \chi(T), and the
NMR relaxation rates, 1/(T_{1}T), can be understood as the effect of the
critical valence fluctuations of f electrons in Yb ion in a unified way. There
also exist a series of Ce-based heavy fermion metals that exhibit anomalies in
physical quantities, enhancements of the residual resistivity \rho_{0} and the
superconducting critical temperature (T_c) around the pressure where the
valence of Ce sharply changes. Here we review the present status of these
problems both from experimental and theoretical aspects.
