  We extend the class of tempered stable distributions first introduced in
Rosinski 2007. Our new class allows for more structure and more variety of tail
behaviors. We discuss various subclasses and the relation between them. To
characterize the possible tails we give detailed results about finiteness of
various moments. We also give necessary and sufficient conditions for the tails
to be regularly varying. This last part allows us to characterize the domain of
attraction to which a particular tempered stable distribution belongs.
