In this paper, an overview of the current state of the study of the heat capacity of carbon nanotubes (CNTs), both pure and with adsorbed gas impurities, is presented. The structural features of CNTs, which determine their thermal properties, are considered, as well as the possibility of the formation of one-dimensional chains of adsorbates on the outer surface of the bundles. The temperature dependence of the heat capacity of multi-walled carbon nanotubes and bundles of single-walled carbon nanotubes as the quasi-low dimensional systems are analyzed. Ferromagnetic catalyst particles lead for the term C_nuc~T⁻² in the heat capacity of the carbon nanotubes at temperatures below 1 K. Above 1 K, the contribution of electrons and nuclear hyperfine contribution is negligibly small in comparison with the phonon contribution. For MWCNTs with the increase of temperature from 2 to 300 K, a changing behavior of temperature dependence of heat capacity from the 3D through 2D to 1D was observed. Particular attention is focused on studies of the heat capacity of one-dimensional chains of atomic and molecular chains of adsorbates located in the outer grooves of bundles of single-walled carbon nanotubes. Heat capacity of 1D chains of atoms of inert gases Xe, Kr, Ar, Ne, and CH₄ molecules adsorbed in grooves (G) on the outer surface of the c-SWCNT bundles was investigated both theoretically and experimentally. The frequencies of longitudinal phonon modes at the edge of the Brillouin zone were determined. The contributions of phonons and thermal vacancies to the heat capacity of xenon and the contributions of phonons and orientation vibrations to the heat capacity of nitrogen were analyzed. The contribution of orientation vibrations (librations) to the heat capacity of nitrogen chains is significant above 15 K. Comparison of the experimental and theoretical phonon curves of the heat capacity for 1D chains of methane indicates a significant contribution of the rotational motion of CH₄ molecules in the all temperature range of experiment.