  Zappa-Sz\'ep products of semigroups encompass both the self-similar group
actions of Nekrashevych and the quasi-lattice-ordered groups of Nica. We use
Li's construction of semigroup $C^*$-algebras to associate a $C^*$-algebra to
Zappa-Sz\'ep products and give an explicit presentation of the algebra. We then
define a quotient $C^*$-algebra that generalises the Cuntz-Pimsner algebras for
self-similar actions. We indicate how known examples, previously viewed as
distinct classes, fit into our unifying framework. We specifically discuss the
Baumslag-Solitar groups, the binary adding machine, the semigroup
$\mathbb{N}\rtimes\mathbb{N}^\times$, and the $ax+b$-semigroup
$\mathbb{Z}\rtimes\mathbb{Z}^\times$.
