Conformal field theories in various dimensions are rich
mathematical objects encoding a wide variety of physical phenomena. For
example, they describe the so-called second-order phase transitions and
provide a non-perturbative window into quantum gravity via the AdS/CFT
correspondence. The last decade has seen a rapid development of an axiomatic
approach to conformal field theories, known as the conformal bootstrap.
Among other things, this has led to the most precise determination of the
critical exponents of the 3D Ising model currently available. In my talk, I
will review the main ideas underlying the conformal bootstrap, focusing on
the interplay between analytical and numerical approaches.