  This is a survey on factorization theory. We discuss finitely generated
monoids (including affine monoids), primary monoids (including numerical
monoids), power sets with set addition, Krull monoids and their various
generalizations, and the multiplicative monoids of domains (including Krull
domains, rings of integer-valued polynomials, orders in algebraic number
fields) and of their ideals. We offer examples for all these classes of monoids
and discuss their main arithmetical finiteness properties. These describe the
structure of their sets of lengths, of the unions of sets of lengths, and their
catenary degrees. We also provide examples where these finiteness properties do
not hold.
