For each 0<e≤1, we establish an upper bound on the anti-canonical volumes of e-lc Fano threefolds. This bound attains the optimal order in e as e approaches 0. Furthermore, relying on the birational models constructed in our earlier work, we prove that the anti-canonical volume of any canonical weak Fano threefold is bounded above by 72, and this bound is sharp. In particular, this settles Prokhorov's conjecture. This talk is based on joint work with Chen Jiang, as well as further joint work with Chen Jiang and Tianqi Zhang.
