variational integrators are a class of geometric structure-preserving numerical integrators for variational differential equations that are based on a discretization of hamilton’s variational principle. we will present our construction of geometric variational integrators for multisymplectic lagrangian and hamiltonian pdes, as well as our construction of geometric variational integrators for adjoint systems arising in optimization and optimal control. for multisymplectic pdes, we will examine their continuous multisymplectic structures and variational principles, and utilize these to develop a discrete variational principle, leading to variational integrators which preserve their multisymplectic structure as well as satisfy a discrete noether’s theorem. for adjoint systems, we will introduce novel type ii variational principles for adjoint systems on vector spaces and lie groups. we utilize these to develop a discrete variational principle for adjoint systems, leading to symplectic and presymplectic integrators for adjoint systems which allow one to compute exact gradients of cost functions in optimization and optimal control problems. numerical examples will be presented throughout.