We develop the concept and the calculus of anti-self dual Lagrangians which seem inherent to many questions in mathematical physics, geometry, and differential equations. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are still minima of action functionals, but just like the self (and antiself) dual equations of quantum field theory (e.g. YangMills) the equations associated to such minima are not derived from the fact they are critical points of the functionals, but because they are also zeroes of the Lagrangian itself. Applications include Navier-Stokes evolutions, various gradient flows, Hamiltonian systems and Cauchy-Riemann equations.