We investigate electronic transport in Josephson junctions formed by individual single-walled carbon nanotubes coupled to superconducting electrodes. We observe enhanced zero-bias conductance (up to 10e 2/h) and pronounced sub-harmonic gap structures in differential conductance, which arise from the multiple Andreev reflections at superconductor/nanotube interfaces. The voltage-current characteristics of these junctions display abrupt switching from the supercurrent branch to the resistive branch, with a gate-tunable switching current ranging from 65 pA to 2.5 nA. The finite resistance observed on the supercurrent branch and the magnitude of the switching current are in good agreement with the classical phase diffusion model for resistively and capacitively shunted junctions.