Appears in collection : 100 (102!) Years of the Ising Model

I will discuss exact solvability results (in a sense) for scaling limits of interface crossings in critical random-cluster models in the plane with various boundary conditions. The results are rigorous for the FK-Ising model, Bernoulli percolation, and the spin-Ising model in appropriate setups. The scaling limit formulas describe structures in the corresponding boundary conformal field theory. (Based on joint works with Yu Feng, Mingchang Liu, and Hao Wu - all at Tsinghua University, China).