Yes, Sherlock Holmes did catch Professor Moriarty, but the encounter led to a dramatic conclusion. In Sir Arthur Conan Doyle’s story “The Final Problem,” Holmes and Moriarty face off at the Reichenbach Falls in Switzerland. After a fierce struggle, both men fall into the waterfall. While Holmes is presumed dead, he later reveals that he survived the fall, but Moriarty did not.
This confrontation marked the end of Moriarty’s criminal empire and his threat to Holmes. However, it also symbolized the high cost of Holmes’s dedication to justice, as he had to fake his own death to escape Moriarty’s network of associates.