  We show that a generic framework $(G,p)$ on the cylinder is globally rigid if
and only if $G$ is a complete graph on at most four vertices or $G$ is both
redundantly rigid and $2$-connected. To prove the theorem we also derive a new
recursive construction of circuits in the simple $(2,2)$-sparse matroid, and a
characterisation of rigidity for generic frameworks on the cylinder when a
single designated vertex is allowed to move off the cylinder.
