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It has long been appreciated that the transport properties of molecules can control reaction kinetics. This effect can be characterized by the time it takes a diffusing molecule to reach a target—the first-passage time (FPT). Determining the FPT distribution in realistic confined geometries has until now, however, seemed intractable. Here, we calculate this FPT distribution analytically and show that transport processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes. Beyond the theoretical aspect, this result changes our views on standard reaction kinetics and we introduce the concept of ‘geometry-controlled kinetics’. More precisely, we argue that geometry—and in particular the initial distance between reactants in ‘compact’ systems—can become a key parameter. These findings could help explain the crucial role that the spatial …