Abstract: We provide a new type of algorithm for discretizing saddle point problems, which implement the inexact algorithms at the continuous level as a multilevel algorithm. A discrete stability condition might not be satisfied and a posteriori error estimates are required only for solutions of symmetric and positive definite problems. The convergence result for the algorithm at the continuous level, combined with standard techniques of discretization and a posteriori error estimates leads to new and efficient algorithms for solving saddle point systems. We introduce the ``Saddle Point Least Squares'' method and relate it with the Brammle-Pasciak's least square method. Numerical results supporting the efficiency of the algorithms are presented for Stokes system, a div-curl system, and for the Maxwell equations. This is joint work with Peter Monk, Francisco Sayas, and Lu Shu.