This work is concerned with the exploration of stable-like random walks on certain non-commutative groups, in particular, nilpotent groups and groups of polynomial volume growth. The word ``exploration'' is used here for two reasons: there is no standard definition of what ``stable-like'' means and, consequently, we will focus on providing classes of natural examples in search of a more formal theory. Taking a wider viewpoint, we focus on the question of understanding what features of a particular driving probability measure determine the basic behavior of the associated random walk.