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We discuss models to describe carrier transport in axial and lateral type carbon nanotube field-effect transistors (CNT-FET). Operation is controlled by the electric field from the gate contact which can lead to strong band bending, allowing carriers to tunnel through the interface barrier. We find that the difference between lateral and axial CNT-FETs is that in devices with axially aligned carbon nanotubes tunneling becomes negligible and transport can be modeled by means of thermionic emission. In lateral CNT-FETs tunneling dominates, for which we present a model for the transmission coefficient using the WKB method and a non-parabolic dispersion relation. The simulated output and transfer characteristics show reasonable agreement with experimental data for both lateral and axial CNT-FET devices.