  We prove $L^{p}$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous
groups, which is one of most general subclasses of nilpotent Lie groups, all
with sharp constants. We also discuss some of their consequences. Already in
the abelian case of $\mathbb{R}^{n}$ our results provide new insights in view
of the arbitrariness of the choice of the not necessarily Euclidean quasi-norm.
