  We derive general properties of anomalous diffusion and nonexponential
relaxation from the theory of tempered \alpha-stable processes. Its most
important application is to overcome the infinite-moment difficulty for the
\alpha-stable random operational time \tau. The tempering results in the
existence of all moments of \tau. The subordination by the inverse tempered
\alpha-stable process provides diffusion(relaxation) that occupies an
intermediate place between subdiffusion (Cole-Cole law) and normal diffusion
(exponential law). Here we obtain explicitly the Fokker-Planck equation, the
mean square displacement and the relaxation function. This model includes
subdiffusion as a particular case.
