I will describe the identification of a supersymmetry-protected subalgebra of the operator algebra of any four-dimensional N=2 SCFT or six-dimensional (2,0) SCFT that takes the form of a two-dimensional chiral algebra. The structure of this protected chiral algebra reflects the physics of the parent four- or six-dimensional theory in interesting ways. In many cases the relevant chiral algebra can be given an economical description as a W-algebra. Applications of this structure include the derivation of new unitarity bounds for four-dimensional SCFTs with global symmetries and the computation of three-point functions of half-BPS operators in the (2,0) theories at finite central charge.