We study the electronic transport properties of dual-gated bilayer graphene devices. We focus on the regime of low temperatures and high electric displacement fields, where we observe a clear exponential dependence of the resistance as a function of displacement field and density, accompanied by a strong nonlinear behavior in the transport characteristics. The effective transport gap is typically 2 orders of magnitude smaller than the optical band gaps reported by infrared spectroscopy studies. Detailed temperature dependence measurements shed light on the different transport mechanisms in different temperature regimes.