We present the Berkeley Crossword Solver, a state-of-the-art approach for automatically solving crossword puzzles. Our system works by generating answer candidates for each crossword clue using neural question answering models and then combines loopy belief propagation with local search to find full puzzle solutions. Compared to existing approaches, our system improves exact puzzle accuracy from 71% to 82% on crosswords from The New York Times and obtains 99.9% letter accuracy on themeless puzzles. Additionally, in 2021, a hybrid of our system and the existing Dr.Fill system outperformed all human competitors for the first time at the American Crossword Puzzle Tournament. To facilitate research on question answering and crossword solving, we analyze our system's remaining errors and release a dataset of over six million question-answer pairs.