  Over the past few years it has been discovered that an "observable" can be
set up on the lattice which obeys the discrete Cauchy-Riemann equations. The
ensuing condition of discrete holomorphicity leads to a system of linear
equations which can be solved to yield the Boltzmann weights of the underlying
lattice model. Surprisingly, these are the well known Boltzmann weights which
satisfy the star-triangle or Yang-Baxter equations at criticality. This
connection has been observed for a number of exactly solved models. I briefly
review these developments and discuss how this connection can be made explicit
in the context of the Z_N model. I also discuss how discrete holomorphicity has
been used in recent breakthroughs in the rigorous proof of some key results in
the theory of planar self-avoiding walks.
