  Let $G$ be a finite group. A finite unordered sequence $S = g_1
\boldsymbol{\cdot} \ldots \boldsymbol{\cdot} g_{\ell}$ of terms from $G$, where
repetition is allowed, is a product-one sequence if its terms can be ordered
such that their product equals $1_G$, the identity element of the group. As
usual, we consider sequences as elements of the free abelian monoid $\mathcal F
(G)$ with basis $G$, and we study the submonoid $\mathcal B (G) \subset
\mathcal F (G)$ of all product-one sequences. This is a finitely generated
C-monoid, which is a Krull monoid if and only if $G$ is abelian. In case of
abelian groups, $\mathcal B (G)$ is a well-studied object. In the present paper
we focus on non-abelian groups, and we study the class semigroup and the
arithmetic of $\mathcal B (G)$.
