We review the cornerstones of the algebraic approach to quantum field theory highlighting its potential effectiveness in addressing concrete open problems in theoretical physics. As an example we discuss the construction of Hadamard states for free quantum fields on four-dimensional asymptotically flat spacetimes by exploiting invariance under the action of the BMS asymptotic symmetry group. In particular we outline how this procedure has been used to give a rigorous construction of the Unruh state for a massless, conformally coupled, real scalar field on Schwarzschild spacetime.
