Conformal blocks are objects of fundamental importance in the bootstrap approach for exact solvability of 2D conformal field theory (CFT).  In this talk, we will present novel probabilistic expressions for them using the Gaussian free field in lieu of their typical construction via representations of the Virasoro algebra.  We will then explain how to apply these expressions to derive analytic properties of conformal blocks such as modular transformations. Our methods are based on recent developments in the probabilistic construction of Liouville CFT, a theory first introduced by Polyakov to describe random surfaces in string theory.   This talk is based on joint work with Promit Ghosal, Guillaume Remy, and Xin Sun.