  In this paper, we establish the core of singular integral theory and
pseudodifferential calculus over the archetypal algebras of noncommutative
geometry: quantum forms of Euclidean spaces and tori. Our results go beyond
Connes' pseudodifferential calculus for rotation algebras, thanks to a new form
of Calder\'on-Zygmund theory over these spaces which crucially incorporates
nonconvolution kernels. We deduce $L_p$-boundedness and Sobolev $p$-estimates
for regular, exotic and forbidden symbols in the expected ranks. In the $L_2$
level both Calder\'on-Vaillancourt and Bourdaud theorems for exotic and
forbidden symbols are also generalized to the quantum setting. As a basic
application of our methods, we prove $L_p$-regularity of solutions for elliptic
PDEs.
