Conformal field theories (CFTs) are an important subclass of quantum field theories which offer a physical description of systems ranging from second-order phase transitions (like water at its critical point) to quantum gravity (through the AdS/CFT correspondence). CFTs come equipped with stringent consistency conditions which - surprisingly - can be used to strongly constrain, if not effectively determine, their observables. We will review this so-called 'conformal bootstrap' approach, highlight some recent results, and discuss future challenges. The second part of the talk will cover a variety of spinoff research directions, including applications to non-perturbative scattering amplitudes.
