  We study large-amplitude oscillations of carbon nanotubes with chiralities
$(m,0)$ and $(m,m)$ and predict the existence of localized nonlinear modes in
the form of {\em discrete breathers}. In nanotubes with the index $(m,0)$ {\em
three types} of localized modes can exist, namely longitudinal, radial, and
twisting breathers; however only the twisting breathers, or {\em twistons}, are
nonradiating nonlinear modes which exist in the frequency gaps of the linear
spectrum. Geometry of carbon nanotubes with the index $(m,m)$ allows only the
existence of broad radial breathers in a narrow spectral range.
