The reflection operators are one of the simplest examples of the non-local integrals of motion in an integrable 2D CFT. They have many interesting applications, including for the description of the UV behavior of massive integrable perturbations of the CFT. In this talk, we’ll discuss the reflection operator in the context of the integrable structure that appears in the study of the quantum KdV theory. This operator is closely related to the Liouville reflection S-matrix. Briefly mentioned will be a new result concerning the full spectrum of this reflection operator, which was obtained recently within the framework of the ODE/IQFT correspondence. Based on Joint work with Gleb Kotousov.