The theory will be developed for the abstract class of multisymplectic PDEs introduced in ( 1.26 ) and repeated here: \(\displaystyle \mathbf {M}Z_t + \mathbf {K}Z_x + \mathbf {N}Z_y = \nabla S(Z)\,,\quad Z\in \mathbb {R}^N\,, \) where \(\mathbf {M}\) , \(\mathbf {K}\) and \(\mathbf {N}\) are \(N\times N\) skew-symmetric matrices and \(\nabla S\) is the gradient of smooth function \(S:\mathbb {R}^N\to \mathbb {R}\) . In this chapter we show various ways that this class of PDEs can arise, both from transforming Hamiltonian and Lagrangian formulations of well-known PDEs, and from pure geometry.