This thesis studies stable-like random walks on finitely generated groups, with particular emphasis on nilpotent groups and, more generally, groups of polynomial growth. It investigates how the driving measure interacts with the algebraic and geometric structure of the underlying group to determine the long-time behavior of the walk. It introduces several natural families of stable-like measures and provided estimates for the associated probabilistic quantities such as probabilities of return. The analysis further elucidates structural relationships among these families and uses them to identify abstract conditions under which stable-like random walks exhibit the desired decay.