The critical point of a two-dimensional statistical model can be often characterized in terms of extended objects that, in the continuum limit, are described by conformal invariant fractal structures The study of these fractals provided new insights into the nature of critical phenomena. In particular, it paved the way to mathematically rigorous approaches and generated a cross-fertilisation line of research between physics and mathematics, between probability and conformal field theory. In this seminar we will focus on critical percolation and its generalizations, and we will outline the questions that arise from the physics point of view. This will lead us to introduce the idea of conformal boostrap in quantum field theory. New results will be presented.