  We develop efficient methods for simulating processes of Ornstein-Uhlenbeck
type related to the class of $p$-tempered $\alpha$-stable ($\ts$)
distributions. Our results hold for both the univariate and multivariate cases
and we consider both the case where the $\ts$ distribution is the stationary
law and where it is the distribution of the background driving L\'evy process
(BDLP). In the latter case, we also derive an explicit representation for the
transition law as this was previous known only in certain special cases and
only for $p=1$ and $\alpha\in[0,1)$. Simulation results suggest that our
methods work well in practice.
