In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievement is probably the proof of conformal invariance of percolation. This theorem, together with the introduction of the so-called Schramm-Loewner Evolution and techniques developed over the years in percolation, allow to describe the critical and near-critical regime of percolation very precisely. These are lecture notes from a course on that topic given at the 2011 "La Pietra week in Probability" in Florence, Italy.