We study the kink modes of oscillation of a curved coronal loop using a toroidal model with a power-law density profile. We find that there are two kink modes and that their velocity polarization is either mainly horizontal or mainly vertical with respect to the photosphere. It is found that the damping by resonant absorption in an inhomogeneous layer between the loop and the coronal environment is slightly more efficient than for the straight magnetic cylinder and that the period and damping time of the two kink modes are very similar. In an equilibrium with nonuniform Alfven frequency, the kink modes display two additional features, namely, the coupling to an Alfven mode of the external medium (which gives rise to resonant absorption in the corona) and wave leakage by tunneling at some distance from the loop. Hence, in general the modes are resonantly damped and leaky at the same time, although our results point out that resonant absorption at the loop boundary is the dominant damping mechanism.
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