Mapping atoms across chemical reactions is important for substructure searches, automatic extraction of reaction rules, identification of metabolic pathways, and more. Unfortunately, the existing mapping algorithms can deal adequately only with relatively simple reactions but not those in which expert chemists would benefit from computer's help. Here we report how a combination of algorithmics and expert chemical knowledge significantly improves the performance of atom mapping, allowing the machine to deal with even the most mechanistically complex chemical and biochemical transformations. The key feature of our approach is the use of few but judiciously chosen reaction templates that are used to generate plausible "intermediate" atom assignments which then guide a graph-theoretical algorithm towards the chemically correct isomorphic mappings. The algorithm performs significantly better than the available state-of-the-art reaction mappers, suggesting its uses in database curation, mechanism assignments, and - above all - machine extraction of reaction rules underlying modern synthesis-planning programs.