  We discuss Monge-Amp\`ere equations from the view point of differential
geometry. It is known that a Monge-Amp\`ere equation corresponds to a special
exterior differential system on a 1-jet space. In this paper, we generalize
Monge-Amp\`ere equations and prove that a $(k+1)$st order generalized
Monge-Amp\`ere equation corresponds to a special exterior differential system
on a $k$-jet space. Then its solution naturally corresponds to an integral
manifold of the corresponding exterior differential system. Moreover, we verify
that the Korteweg-de Vries (KdV) equation and the Cauchy-Riemann equations are
examples of our equation.
