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-Bab\v uska-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.