  We describe a general procedure to produce fundamental domains for complex
hyperbolic triangle groups, a class of groups that contains a representative of
the commensurability class of every known non-arithmetic lattice in ${\rm
PU}(2,1)$. We discuss several commensurability invariants for lattices, and
show that some triangle groups yield new commensurability classes, bringing the
number of known non-arithmetic commensurability classes to 22.
