N=4 Super Yang Mills theory has a large set of interesting half BPS, conformally invariant boundary conditions. S-duality acts in surprising ways on that set, mapping even elementary boundary conditions, as Neumann or Dirichlet, to rather intricate ones. We will give detailed examples of S-dual pairs of boundary conditions for U(1), SU(2) gauge groups, and introduce a general recipe for any gauge group. Besides the physical interest, the subject has various mathematical applications, including symplectic duality (aka 3d mirror symmetry) and geometric Langlands, and can be formulated as an interesting mathematical problem in its own sake.