  After a brief review of the historical role of analyticity in the study of
critical phenomena, an account is given of recent discoveries of discretely
holomorphic observables in critical two-dimensional lattice models. These are
objects whose correlation functions satisfy a discrete version of the
Cauchy-Riemann relations. Their existence appears to have a deep relation with
the integrability of the model, and they are presumably the lattice versions of
the truly holomorphic observables appearing in the conformal field theory (CFT)
describing the continuum limit. This hypothesis sheds light on the connection
between CFT and integrability, and, if verified, can also be used to prove that
the scaling limit of certain discrete curves in these models is described by
Schramm-Loewner evolution (SLE).
