Tempered stable distributions were introduced in Rosin?ski 2007 as models that look like infinite variance stable distributions in 
 some central region, but they have lighter (i.e. tempered) tails. We introduce a larger class of models that allow for more variety in 
 the tails. While some cases no longer correspond to stable distributions they serve to make the class more flexible, and in certain 
 subclasses they have been shown to provide a good fit to data. To characterize the possible tails we give detailed results about finite
- ness 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. We will also characterize the weak 
 limits of sequences of tempered stable distributions. If time permits, we will motivate why distributions that are stable-like in some
 central region but with lighter tails may show up in applications.