In this talk, we consider p and h-p least-squares spectral element methods for elliptic boundary layer problems in one dimension. We derive stability estimates and design a numerical scheme based on minimizing the residuals in the sense of least-squares in appropriate Sobolev norms. We prove parameter robust uniform error estimates i.e. error in the approximation is independent of the boundary layer parameter for the p and hp-version. Numerical results are presented for a number of model elliptic boundary layer problems confirming the theoretical estimates and uniform convergence results.