Abstract: The space of Virasoro conformal blocks on a Riemann surface is a central object in conformal field theory, studied from many different points of view. I will recall what this space is and what some of its expected structures are, and then describe a new method for constructing conformal blocks, when the Virasoro central charge is c=1. Potential applications include new formulas for isomonodromic tau-functions, and the proof of a conjecture of Goncharov-Shen relating conformal blocks to quantization of cluster varieties. This is a report of joint work with Qianyu Hao and work in progress with Dawit Belayneh and Davide Gaiotto.