How does the carbon fusion reaction happen in stars?

The 12C + 12C fusion reaction is one of the most important reactions in the stellar evolution. Due to its compli-cated reaction mechanism, there is great uncertainty in the reaction rate which limits our understanding of vari-ous stellar objects, such as explosions on the surface of neutron stars, white dwarf (type Ia) supernovae, and massive stellar evolution. In this paper, I will review the challenges in the study of carbon burning. I will also report recent re-sults from our studies: 1) an upper limit for the 12C + 12C fusion cross sections, 2) measurement of the 12C + 12C at deep sub-barrier energies, 3) a new measurement of the 12C(12C, n) reaction. The outlook for the studies of the astrophysical heavy-ion fusion reactions will also be presented.