We consider differential-geometric structures associated with Monge–Amp`ere equations on manifolds and use them to study the con- tact linearization of such equations. We also consider the category of Monge–Amp`ere equations (the morphisms are contact diffeomorphisms) and a number of subcategories. We are chiefly interested in subcategories of Monge–Amp`ere equations whose objects are locally contact equivalent to equations linear in the second derivatives (semilinear equations), linear in derivatives, almost linear, linear in the second derivatives and inde- pendent of the first derivatives, linear, linear and independent of the first derivatives, equations with constant coefficients or evolution equations. We construct a number of functors from the category of Monge–Amp`ere equa- tions and from some of its subcategories to the category of tensor objects (that is, multi-valued sections of tensor bundles). In particular, we con- struct a pseudo-Riemannian metric for every generic Monge–Amp`ere equa- tion. These functors enable us to establish effectively verifiable criteria for a Monge–Amp`ere equation to belong to the subcategories listed above.