Conformal invariance for critical Ising model

We discuss the critical 2D Ising model, in particular various conjectures concerning its conformal invariance. A number of these conjectures were recently proven by S. Smirnov and his school, including D. Chelkak, C. Hongler, A. Kemppainen, K. Kytola and the speaker. The key role in the proof of conformal invariance is played by discrete holomorphic fermionic observables. We will briefly discuss the proof of their convergence, as well as applications, namely, convergence of interfaces to the SLE curves, convergence of the energy field and some partial results concerning the spin field.