  Oftentimes the elements of a ring or semigroup $H$ can be written as finite
products of irreducible elements, say $a=u_1 \cdot \ldots \cdot u_k = v_1 \cdot
\ldots \cdot v_{\ell}$, where the number of irreducible factors is distinct.
The set $\mathsf L (a) \subset \mathbb N$ of all possible factorization lengths
of $a$ is called the set of lengths of $a$, and the full system $\mathcal L (H)
= \{ \mathsf L (a) \mid a \in H \}$ is a well-studied means of describing the
non-uniqueness of factorizations of $H$. We provide a friendly introduction,
which is largely self-contained, to what is known about systems of sets of
lengths for rings of integers of algebraic number fields and for transfer Krull
monoids of finite type as their generalization.
