Are there bounds on quantum information processing and thermalization in many-body quantum systems? The fast-scrambling conjecture states that quantum information processing times must grow at least logarithmically with the number of degrees of freedom. We derive this result for generic systems. We discuss the nonuniversal relationship between scrambling and many-body quantum chaos, which is often used as a diagnostic for fast scrambling. We also find explicit examples of sparsely connected quantum systems which are fast scramblers, suggesting additional routes to experimentally testing the fast-scrambling conjecture in cold atomic gases. These results clarify for which information-processing tasks quantum black holes may be the fastest quantum systems.