Geometry of quantized PDE's.

In this paper we resume some recent results in the direction of the formal quantization of PDE's obtained by the author, and also announce some new further results. The categorial meaning of quantization of PDE's is given. Formal quantization results a canonical functor defined on the category of differential equations. Furthermore, a Dirac-quantization can be interpreted as a covering in the category of differential equations. A quantum (pre-)spectral measure is a functor that can be factorized by means of formal quantization and a (pre-)spectral measure. A relation between canonical Dirac-quantization and singular solutions of PDE's is given. It is proved also that knowledge of B\"acklund correspondeces, as well conservation laws, can aid the proceeding of canonical quantization of PDE's.