We propose, for the first time, optimal design for changepoint problems. Suppose that a sequence of observations is taken in some subinterval of the real axis. If the distribution of the sequence changes at some unknown location then we refer to this location as a changepoint. Changepoint inference usually concerns location testing for a change and/or estimating the location of the change and the unknown parameters of the distributions before and after any change. In this paper, we investigate Bayesian optimal designs for changepoint problems. We find robust optimal designs which allow for arbitrary distributions before and after the change, arbitrary prior densities on the parameters before and after the change, and any log-concave prior density on the changepoint. We define a new design measure for Bayesian optimal design problems as a means of finding the optimal design itself. Our results apply to any design criterion function concave in the design measure. We show that our method extends directly to a setting in which there are several paths all with the same changepoint.