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We consider a natural class of long range random walks on torsion freenilpotent groups and develop limit theorems for these walks. Given the originaldiscrete group $\Gamma$ and a random walk $(S_n)_ {n\ge1}$ driven by a certaintype of symmetric probability measure $\mu$, we construct a homogeneousnilpotent Lie group $G_\bullet(\Gamma,\mu)$ which carries an adapted dilationstructure and a stable-like process $(X_t)_{ t\ge0}$ which appears in aDonsker-type functional limit theorem as the limit of a rescaled version of therandom walk. Both the limit group and the limit process on that group depend onthe measure $\mu$. In addition, the functional limit theorem is complemented bya local limit theorem.

Limit theorems for some long range random walks on torsion free nilpotent groups