Fusion reactions not only provide key information for studying the dynamic evolution and dissipation mechanisms in quantum many-body systems but also open up an important avenue for exploring the reaction dynamics and structural characteristics of atomic nuclei. In recent years, with the continuous development of the technology for synthesizing new elements and their isotopes via fusion reactions, a series of new elements and their isotopes has been successfully synthesized. In this paper, the synthesis pathways of elements in different mass regions, ranging from hydrogen to mendelevium, as well as the experimental progress of various heavy-ion fusion reactions from light systems to heavy systems are systematically summarized, and the advantages and limitations of current theoretical models in describing the capture process, and focuses on analyzing the strengths and shortcomings of phenomenological models and microscopic dynamic models in explaining the fusion behavior of different reaction systems are also reviewed. For the capture cross sections in light nuclei-light nuclei reaction systems, the EBD method, the CCFULL model, the universal Wong formula, and the ImQMD model all demonstrate good agreement with the experimental data. For the systems involving light nuclei-medium mass nuclei and light nuclei-heavy nuclei, the above-mentioned models provide satisfactory descriptions. In particular, for the ¹⁶O+¹⁴⁴Sm reaction system, the results obtained from the CCFULL model show good agreement with experimental data across both the sub-barrier and above-barrier energy regions. For the heavy nuclei-heavy nuclei systems, however, the EBD method holds a distinct advantage. Therefore, in subsequent predictions of the evaporation residue cross sections for superheavy elements, the results calculated by the EBD method can serve as the input for the capture cross section. On this basis, several key scientific issues in fusion reaction research are proposed, including heavy-ion fusion hindrance, the phenomenon of fusion suppression at extreme sub-barrier energies, fusion probability P_\textCN, and the fission barrier of compound nuclei. Furthermore, an outlook and suggestions for future research directions in fusion reactions are provided. The datasets presented in this paper are openly available at https://doi.org/10.57760/sciencedb.j00213.00238.