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Abstract: Living things enjoy exquisite molecular sensitivity in their key processes, including DNA replication, transcription and translation, sensing, and morphogenesis. Understanding the diverse chemical mechanisms responsible for this sensitivity poses a basic challenge for nonequilibrium physics. At equilibrium, strict constraints relate sensitivity to system structure and spontaneous fluctuations. But away from equilibrium, we lack knowledge of even the most basic limits to sensitivity set by coarse properties like system size or dissipation. My colleagues and I have uncovered simple laws that constrain sensitivity in small, nonequilibrium systems. Our results extend and unify a patchwork of prior biophysics results on the energetic and structural costs of sharp biochemical switches, accurate sensors, and molecular discrimination---revealing their common origin in the perturbation theory of Markov chains. In pursuit of mechanisms that saturate the bounds we have uncovered, we discover a nonequilibrium binding mechanism with remarkable sensitivity, exponential in system size, with implications for our understanding of models of gene regulation.