Ising Model and Conformal Field Theory: a rigorous connection

The Ising model is perhaps the most classical model of equilibrium statistical mechanics. Thanks to exactly solvable structures, it is amenable to a very precise analysis, which has led to many spectacular results over the last 70 years.
In the 1970s, a connection between the critical Ising model with Quantum Field Theory was conjectured by Kadanoff and others. In the 1980s this conjecture was refined into a connection between the planar Ising model and the first unitary minimal model of CFT, which led to (conjectural) exact formulae for the correlation functions, including in finite geometries.
In the early 2000s, mathematicians introduced new tools to study conformal symmetry, in particular discrete complex analysis and the Schramm's SLE random curves.
I will discuss how in the recent years we were able to prove, using discrete complex analysis and probabilistic techniques, that the CFT formulae indeed describe the scaling limit of the 2D Ising model on arbitrary geometries, how the random curves of the models can be described in terms of SLE and how we are now in position to rigorously connect, in a conceptually satisfactory manner, the Ising model and its Conformal Field Theory.