  In this paper we give an overview of exactly solved edge-interaction models,
where the spins are placed on sites of a planar lattice and interact through
edges connecting the sites. We only consider the case of a single spin degree
of freedom at each site of the lattice. The Yang-Baxter equation for such
models takes a particular simple form called the star-triangle relation.
Interestigly all known solutions of this relation can be obtained as particular
cases of a single "master solution", which is expressed through the elliptic
gamma function and have continuous spins taking values of the circle. We show
that in the low-temperature (or quasi-classical) limit these lattice models
reproduce classical discrete integrable systems on planar graphs previously
obtained and classified by Adler, Bobenko and Suris through the
consistency-around-a-cube approach. We also discuss inversion relations, the
physicical meaning of Baxter's rapidity-independent parameter in the
star-triangle relations and the invariance of the action of the classical
systems under the star-triangle (or cube-flip) transformation of the lattice,
which is a direct consequence of Baxter's Z-invariance in the associated
lattice models.
