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