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We derive general properties of anomalous diffusion and nonexponential relaxation from the theory of tempered -stable processes. The tempering results in the existence of all moments of operational time. The subordination by the inverse tempered -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 and the Cole-Davidson relaxation function. This model includes subdiffusion as a particular case.