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We present a dynamic resource allocation strategy for energy-efficient and Electromagnetic Field (EMF) exposure aware computation offloading at the wireless network edge. The goal is to maximize the overall system sum-rate of offloaded data, under stability (i.e. finite end-to-end delay), EMF exposure and system power constraints. The latter comprises end devices for uplink transmission and a Mobile Edge Host (MEH) for computation. Our proposed method, based on Lyapunov stochastic optimization, is able to achieve this goal with theoretical guarantees on asymptotic optimality, without any prior knowledge of wireless channel statistics. Although a complex long-term optimization problem is formulated, a per-slot optimization based on instantaneous realizations is derived. Moreover, the solution of the instantaneous problem is provided with closed form expressions and fast iterative procedures. Besides the theoretical analysis, numerical results assess the performance of the proposed strategy in striking the best trade-off between offloading sum-rate, power consumption, EMF exposure, and E2E delay. To the best of our knowledge, this is the first work addressing the problem of energy and exposure aware computation offloading.