  Conformal invariance plays a significant role in many areas of Physics, such
as conformal field theory, renormalization theory, turbulence, general
relativity. Naturally, it also plays an important role in geometry: theory of
Riemannian surfaces, Weyl tensors, $Q$-curvature, Yang-Mills fields, etc... We
shall be concerned with the study of conformal invariance in analysis. More
precisely, we will focus on the study of nonlinear PDEs arising from
conformally invariant two dimensional variational problems (e.g. harmonic maps,
prescribed mean curvature surfaces, Willmore and Constrained conformal
surfaces, isothermic surfaces). The present manuscript are lecture notes of
courses given by the author at several places including UBC Vancouver, SNS
Pisa, IHP Paris, ICTP Trieste.
