  We extend a few fundamental aspects of the classical theory of non-unique
factorization, as presented in Geroldinger and Halter-Koch's 2006 monograph on
the subject, to a non-commutative and non-cancellative setting, in the same
spirit of Baeth and Smertnig's work on the factorization theory of
non-commutative, but cancellative monoids [J. Algebra 441 (2015), 475-551].
Then, we bring in power monoids and, applying the abstract machinery developed
in the first part, we undertake the study of their arithmetic.
  More in particular, let $H$ be a multiplicatively written monoid. The set
$\mathcal P_{\rm fin}(H)$ of all non-empty finite subsets of $H$ is naturally
made into a monoid, which we call the power monoid of $H$ and is
non-cancellative unless $H$ is trivial, by endowing it with the operation
$(X,Y) \mapsto \{xy: (x,y) \in X \times Y\}$. Power monoids are, in disguise,
one of the primary objects of interest in arithmetic combinatorics, and here
for the first time we tackle them from the perspective of factorization theory.
Proofs lead to consider various properties of finite subsets of $\mathbf N$
that can or cannot be split into a sumset in a non-trivial way, which gives
rise to a rich interplay with additive number theory.
