This paper reviews recent research studies on the buckling of carbon nanotubes. The structure and properties of carbon nanotubes are introduced to the readers. The various buckling behaviors exhibited by carbon nanotubes are also presented herein. The main factors, such as dimensions, boundary conditions, temperature, strain rate, and chirality, influencing the buckling behaviors are also discussed, as well as a brief introduction of the two most used methods for analyzing carbon nanotubes, i.e., continuum models and atomistic simulations. Summary and recommendations for future research are also given. Finally, a large body of papers is given in the reference section. It is hoped that this paper provides current knowledge on the buckling of carbon nanotubes, reviews the computational methods for determining the buckling loads, and inspires researchers to further investigate the buckling properties of carbon nanotubes for practical applications.