We establish distributional limit theorems for the shape statistics of a concave majorant (i.e. the fluctuations of its length, its supremum, the time it is attained and its value at $T$) of any Lévy process on $[0,T]$ as $T\to\infty$. The scale of the fluctuations of the length and other statistics, as well as their asymptotic dependence, vary significantly with the tail behaviour of the Lévy measure. The key tool in the proofs is the recent representation of the concave majorant for all Lévy processes using a stick-breaking representation.