  The leitmotiv of this review is noncommutative principal U(1)-bundles and
associated line bundles. In the first part I give a brief introduction to
Hopf-Galois theory and its applications, from field extensions to principal
group actions. I then recall Woronowicz' definition of compact quantum group
and the notion of noncommutative principal bundle. When the structure group is
U(1), there is a construction due to Pimsner that allows to get the total space
of a "bundle" (more precisely, a strongly graded C*-algebra) from the base
space and a noncommutative "line bundle" (a self-Morita equivalence bimodule).
As an example of this construction, I will discuss the U(1)-principal bundles
of quantum lens spaces over quantum weighted projective space. The second part
is a peek into the realm of nonassociative geometry: after a review of some
properties of Hopf cochains and cocycles, I will discuss the theory of cochain
quantization and its applications, from Albuquerque-Majid example of octonions,
to "line bundles" on the noncommutative torus.
