I will review the recent mathematical approach to Conformal Field Theory proposed by my colleagues Nam-Gyu Kang and Nikolai Makarov. Their construction defines a certain class of algebraic operations on correlation functions of the Gaussian Free Field, and these operations can be used to give meaning to "vertex observables" and other well known objects in CFT. Using conformal transformation rules for the GFF these objects can be defined on any simply connected domain, and using Lie derivatives they can be analyzed when the domain evolves according to an infinitesimal flow. Using the flow of Loewner's differential equation produces a connection with the random curves of the Schramm-Loewner evolution, which I will describe along with some recent work in the case of multiple SLE curves.