The Yang-Baxter and the pentagon equation serve as important
equations in mathematical physics. They appear in two equally
significant versions, the operator and the set-theoretical one. In this
talk, we focus on the set-theoretic versions of both equations, where
their solutions are known as Yang-Baxter maps and pentagon maps,
respectively. First, we recall rational Yang-Baxter maps of a specific
type (quadrirational maps) and show their connection to discrete
integrable systems. Then, we propose a classification scheme for
quadrirational solutions of the pentagon equation. That is, we give a
full list of representatives of quadrirational maps that satisfy the
pentagon equation, modulo an equivalence relation that is defined on
birational functions on $\mathbb{CP}^1 \times \mathbb{CP}^1$. Finally,
we demonstrate how Yang-Baxter maps can be derived from quadrirational
pentagon maps.