  Fractional tempered stable motion (fTSm)} is defined and studied. FTSm has
the same covariance structure as fractional Brownian motion, while having tails
heavier than Gaussian but lighter than stable. Moreover, in short time it is
close to fractional stable L\'evy motion, while it is approximately fractional
Brownian motion in long time. A series representation of fTSm is derived and
used for simulation and to study some of its sample path properties.
