in this talk, i plan to discuss how differential geometry can provide useful insights into the study of ordinary and partial differential equations. in particular, i will focus on the role of symplectic geometry in classical lagrangian and hamiltonian mechanics, as well as its generalization to the multisymplectic geometry of classical field theory. finally, i will talk about how this perspective has paved the way for the development of ``geometric'' numerical integrators, which exactly preserve important structures, symmetries, and invariants.