We describe an important class of semidefinite programming problems that has received scant attention in the optimization community. These problems are derived from considerations in distance geometry and multidimensional scaling and therefore arise in a variety of disciplines, e.g. computational chemistry and psychometrics. In most applications, the feasible positive semidefinite matrices are restricted in rank, so that recent interior-point methods for semidefinite programming do not apply. We establish some theory for these problems and discuss what remains to be accomplished.