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Projection stabilisation applied to general Lagrange multiplier finiteelement methods is introduced and analysed in an abstract framework. We thenconsider some applications of the stabilised methods: (i) the weak impositionof boundary conditions, (ii) multiphysics coupling on unfitted meshes, (iii) anew interpretation of the classical residual stabilised Lagrange multipliermethod introduced in H. J. C. Barbosa and T. J. R. Hughes, The finite elementmethod with Lagrange multipliers on the boundary: circumventing theBabu\v{s}ka-Brezzi condition, Comput. Methods Appl. Mech. Engrg.,85(1):109--128, 1991 .

Projection stabilisation of Lagrange multipliers for the imposition of constraints on interfaces and boundaries