The basic biochemical functions of life are carried out by large molecules called enzymes. Enzymes consist of long chains of amino acids folded into a three-dimensional structure. Within that structure, a specific cluster of amino acids, known as the active site, performs the biochemical function. Substituting one amino acid for another in the active site typically results in a defective, non-functional enzyme, and therefore mutations at or near enzyme active sites are often lethal. Moreover, even mutations far from the active site have been found to disrupt function. Nonetheless, as organisms evolve, enzymes accumulate random mutations. Where in enzymes' structures do these mutations accumulate without causing harm? We observe evidence for extensive interactions between active sites and distant regions of the enzyme structure, in a comprehensive set of over 500 enzymes. We show that active sites tightly control the substitutions that an enzyme can tolerate. This control extends far beyond regions of the enzyme immediately adjacent to the active site, covering over 80% of a typical enzyme structure. Our findings have broad implications for molecular evolution, for enzyme engineering, and for the computational prediction of active-site locations in novel enzymes.