  Critical points and singularities are encountered in the study of critical
phenomena in probability and physics. We present recent results concerning the
values of such critical points and the nature of the singularities for two
prominent probabilistic models, namely percolation and the more general
random-cluster model. The main topic is the statement and proof of the
criticality and universality of the canonical measure of bond percolation on
isoradial graphs (due to the author and Ioan Manolescu). The key technique used
in this work is the star--triangle transformation, known also as the
Yang--Baxter equation. The second topic reported here is the identification of
the critical point of the random-cluster model on the square lattice (due to
Beffara and Duminil-Copin), and of the criticality of the canonical measure of
the random-cluster model with q \ge 4 on periodic isoradial graphs (by the same
authors with Smirnov). The proof of universality for percolation is expected to
extend to the random-cluster model on isoradial graphs.
