Tempered fractional Brownian motion (TFBM) is defined by exponentially tempering the power law kernel in the moving average representation of a fractional Brownian motion (FBM). TFBM is a Gaussian process with stationary increments, and we call those increments tempered fractional Gaussian noise (TFGN). TFGN exhibits semi-long range dependence. That is, its autocovariance function closely resembles that of fractional Gaussian noise on an intermediate scale, but then it eventually falls off more rapidly. The spectral density of TFGN resembles a negative power law for low frequencies, but eventually converges to zero at very low frequencies. This behavior of the spectral density is consistent with the Davenport spectrum that extends the $5/3$ Kolmogorov theory of turbulence beyond the inertial range. TFBM is a linear combination of tempered fractional integrals (or derivatives) of a white noise. Using that fact, we developed the theory of stochastic integration for TFBM. Replacing the Gaussian random measure in the moving average or harmonizable representation of TFBM by a stable random measure, we obtained a linear tempered fractional stable motion (LTFSM), or a real harmonizable tempered fractional stable motion (HTFSM), respectively.