This book presents, for the first time, a systematic formulation of the geometric theory of noncommutative PDE's which is suitable enough to be used for a mathematical description of quantum dynamics and quantum field theory. A geometric theory of supersymmetric quantum PDE's is also considered, in order to describe quantum supergravity. Covariant and canonical quantizations of (super) PDE's are shown to be founded on the geometric theory of PDE's and to produce quantum (super) PDE's by means of functors from the category of commutative (super) PDE's to the category of quantum (super) PDE's. Global properties of solutions to (super) (commutative) PDE's are obtained by means of their integral bordism groups.Contents: Quantized PDE's I: Noncommutative ManifoldsQuantized PDE's II: Noncommutative PDE'sQuantized PDE's III: Quantizations of Commutative PDE'sAddendum I: Bordism Groups and the (NS)-ProblemAddendum II: Bordism Groups and Variational PDE'sReadership: Researchers and graduate students in the fields of partial differential equations, mathematical physics and theoretical physics.