Starting from an overview of general concepts in Fischer-Tropsch synthesis, this perspective article focuses on relevant theoretical works based on density functional theory (DFT) dealing with cobalt based Fischer-Tropsch (FT) catalysis. Our bibliographic analysis highlights key scientific questions in a challenging field for research and industry. Starting from the historical and empirical concepts, we discuss how recent DFT studies give insights into the catalyst structure, the FT reaction mechanisms and the deactivation pathways. There is still no clear consensus within the scientific community about the FT reaction mechanism. Some authors propose that the first elementary step in the FT synthesis consists on CO dissociation taking place at the steps of the cobalt active phase (carbide mechanism) while some others suggest that this molecule is hydrogenated prior to its dissociation no matter the structure of the active site (insertion mechanism). As for the crucial deactivation problem, according to the computed thermodynamic properties presented by some publications, the catalyst oxidation does not seem to play a major role in its deactivation process. By contrast, the interaction of C addatoms with the Co active phase can have a strong influence on the catalyst reactivity and selectivity, through a series of structural rearrangements: the formation of various more or less unsaturated hydrocarbon molecules (such as polyaromatic rings) and the formation of a surface carbide phase. However, the impact of these phenomena on the reactivity and selectivity of the catalyst remain to be investigated in the future.