One definition of Quantum Field Theory is as an associative algebra of local operators.
Though seemingly formal, this point of view has been extremely successful in the realm
of two spacetime dimensions, where many interesting (and experimentally relevant)
QFTs can be thus constructed without any reference to Lagrangians. On the contrary, until recently Lagrangians remained crucial in most practical QFT computations in three and four dimensions. Enter the recently revived "Conformal bootstrap"...