The purpose of this paper is to develop and analyze least-squares approximations for Stokes and elasticity problems. The major advantage of the least-square formulation is that it does not require that the classical Ladyzhenskaya-Babuska-Brezzi (LBB) condition be satisfied. We provide two methods. The first is posed in terms of the velocity-pressure pair without the introduction of additional variables. The second adds a vorticity variable. In both cases, we employ least-squares functionals which involve a discrete inner product which is related to the inner product in H-1(\d) (the Sobolev space of order minus one on \d). The use of such inner products (applied to second order problems) was proposed in an earlier paper by Bramble, Lazarov, and Pasciak .