Gamma-ray bursts (GRBs) have been an enigma since their discovery forty years ago. However, considerable progress unraveling their mysteries has been made in recent years. Developments in observations, theory, and instrumentation have prepared the way so that the next decade can be the one in which we finally answer the question, What are gamma-ray bursts? This question encompasses not only what the progenitors are that produce the GRBs, but also how the enormous luminosity of the GRBs, concentrated in gamma rays, is achieved. Observations across the electromagnetic spectrum, from both the ground and space, will be required to fully tackle this important question. This white paper, mostly distilled from a recent study commissioned by the Division of Astrophysics of the American Physical Society, focuses on what very high energy (~100 GeV and above) gamma-ray observations can contribute. Very high energy gamma rays probe the most extreme high energy particle populations in the burst environment, testing models of lepton and proton acceleration in GRBs and constraining the bulk Lorentz factor and opacity of the outflow. Sensitivity improvements of more than an order of magnitude in the very high energy gamma-ray band can be achieved early in the next decade, in order to contribute to this science.
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