Abstract: Two different preconditioners for symmetric saddle point problems with a penalty term are analyzed. The saddle point problems are discretized by mixed finite elements. The preconditioners are applied in combination with Krylov space methods. It is shown that both methods yield convergence rates that are independent from both, the discretization and the penalty parameters. The first method is based on a symmetric positive definite block-diagonal preconditioner and the second one uses a non-symmetric and indefinite block-triangular preconditioner. Numerical results are presented for a problem of linear elasticity. The preconditioners in our experiments are based on domain decomposition and multilevel techniques. It is further shown that the analysis of the symmetric positive definite preconditioner can also be applied to construct preconditioners for symmetric indefinite problems arising from second-order elliptic equations. Numerical results are presented for the Helmholtz equation.