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We have comprehensively studied the influence of the channel and underlap engineering on the switching and high-frequency performance of carbon nanotube field-effect transistors (CNTFETs). Various source/drain and channel-engineered CNTFETs have been investigated and optimized architectures have been concluded from simulations. Performance parameters such as switching time τ and cutoff frequency fT as well as ambipolarity and on/off ratio Ion/ Ioff have been calculated and optimized. New CNTFETs with staircase doping in the underlap region (SU-CNTFETs)] have been proposed and the optimized underlap length and doping level have been concluded. Adding this kind of underlaps to conventional CNTFETs improves fT, τ, and Ion/I off and limits the leakage current simultaneously. For the first time, laterally asymmetric channel CNTFETs with a realistic staircase doping (SLAC-CNTFETs) have been proposed. It is shown that SLAC-CNTFET has an improved RF and switching performance while maintaining a good Ion /Ioff. The effect of single halo implantation on the aforementioned parameters has been calculated and it is revealed that it degrades the figures of merit. Detailed reasoning of the aforementioned characteristics has been discussed based on the gate and channel electric field distributions and the quantum capacitance. The impact of fringing field capacitances on fT has been computed and shown that it is not negligible. Finally, our calculations for an individual CNTFET with metallic contacts showed that the Schottky barriers at the contacts deteriorate almost all the figures of merits except Ioff.