  We offer a unified approach to the theory of concave majorants of random
walks by providing a path transformation for a walk of finite length that
leaves the law of the walk unchanged whilst providing complete information
about the concave majorant. This leads to a description of a walk of random
geometric length as a Poisson point process of excursions away from its concave
majorant, which is then used to find a complete description of the concave
majorant for a walk of infinite length. In the case where subsets of increments
may have the same arithmetic mean, we investigate three nested compositions
that naturally arise from our construction of the concave majorant.
