A remarkable development of the last 50 years in physics is the unification of statistical mechanics and quantum field theory, into a field sometimes called Statistical Field Theory. The study of planar lattice models, such as the Ising model, allows one to gain a concrete insight into what this means, bringing together beautiful pieces of mathematics (in particular conformal geometry, complex analysis and probability) to get exciting physical results, by following the path of this unification. 
The goal of this mini-course is to give an idea about how we can start with a lattice model, end up with a conformal field theory, and use the symmetries of the latter to get spectacular formulae about the former.