  We present a framework for performing regression when both covariate and
response are probability distributions on a compact interval
$\Omega\subset\mathbb{R}$. Our regression model is based on the theory of
optimal transportation and links the conditional Fr\'echet mean of the response
distribution to the covariate distribution via an optimal transport map. We
define a Fr\'echet-least-squares estimator of this regression map, and
establish its consistency and rate of convergence to the true map, under both
full and partial observation of the regression pairs. Computation of the
estimator is shown to reduce to an isotonic regression problem, and thus our
regression model can be implemented with ease. We illustrate our methodology
using real and simulated data.
