  A new class of random composition structures (the ordered analog of Kingman's
partition structures) is defined by a regenerative description of component
sizes. Each regenerative composition structure is represented by a process of
random sampling of points from an exponential distribution on the positive
halfline, and separating the points into clusters by an independent
regenerative random set. Examples are composition structures derived from
residual allocation models, including one associated with the Ewens sampling
formula, and composition structures derived from the zero set of a Brownian
motion or Bessel process. We provide characterisation results and formulas
relating the distribution of the regenerative composition to the L{\'e}vy
parameters of a subordinator whose range is the corresponding regenerative set.
In particular, the only reversible regenerative composition structures are
those associated with the interval partition of $[0,1]$ generated by excursions
of a standard Bessel bridge of dimension $2 - 2 \alpha$ for some $\alpha \in
[0,1]$.
