  We consider a new family of $\R^d$-valued L\'{e}vy processes that we call
Lamperti stable. One of the advantages of this class is that the law of many
related functionals can be computed explicitely (see for instance \cite{cc},
\cite{ckp}, \cite{kp} and \cite{pp}). This family of processes shares many
properties with the tempered stable and the layered stable processes, defined
in Rosi\'nski \cite{ro} and Houdr\'e and Kawai \cite{hok} respectively, for
instance their short and long time behaviour. Additionally, in the real valued
case we find a series representation which is used for sample paths simulation.
In this work we find general properties of this class and we also provide many
examples, some of which appear in recent literature.
