I will discuss recent progress on the vanishing-viscosity limit of the two-dimensional Navier–Stokes equation. Our approach is Lagrangian and probabilistic: 1. We develop a stochastic counterpart of the DiPerna–Lions theory to construct and control stochastic Lagrangian flows for the viscous dynamics. 2. We also establish a large-deviation principle that quantifies convergence to the Euler dynamics. This talk is based on joint work with Chanwoo Kim, Dohyun Kwon, and Jinsol Seo.