  This paper is the 2nd part of a two-paper series whose aim is to give a
detailed description of Connes' pseudodifferential calculus on noncommutative
$n$-tori, $n\geq 2$. We make use of the tools introduced in the 1st part to
deal with the main properties of pseudodifferential operators on noncommutative
tori of any dimension $n\geq 2$. This includes the main results mentioned in
the original notes of Connes and Baaj. We also obtain further results regarding
action on Sobolev spaces, spectral theory of elliptic operators, and
Schatten-class properties of pseudodifferential operators of negative order,
including a trace-formula for pseudodifferential operators of order $<-n$.
