The Bergelson conjecture from 1996 asserts that the multilinear polynomial ergodic averages with commuting transformations converge pointwise almost everywhere in any measure-preserving system. This problem was recently solved affirmatively for polynomials with distinct degrees. In this talk, I will review the recent progress on this conjecture, focusing on the multilinear circle method—a versatile new tool that combines methods from additive combinatorics and Fourier analysis, which are crucial in problems of this kind. This is based on joint work with D. Kosz, S. Peluse and J. Wright.