Augmented Lagrangians and Lagrangians are constrained optimization tools that very early have naturally been applied to contact problems with deformable solids. The augmented Lagrangian has since quite widely become established for the approximation and resolution of contact problems in small and large strains. The method by Nitsche was originally proposed to allow a Dirichlet-type boundary condition to be weakly taken into account, precisely avoiding the use of Lagrange multipliers. Only recently has it been extended to contact conditions with or without friction. The close connection between Nitsche and Lagrangian methods is however quite clear and it is the objective of this chapter to shed some light on this relationship. This is achieved namely by looking into the mechanisms underlying these methods, and also by way of presenting some recent developments within the framework of small and large elastic strains.