In 1960, Kinnunen and Nylander presented a mechanical approach for calculating the punching resistance of slab-column connections. That approach accounted for the response of a slab sector and considered failure to occur on the basis of the strains developing in the soffit of the slab. The model constituted a major step forward in the understanding of the punching phenomenon, being the most advanced mechanical theory at that time. Based on similar considerations, the Critical Shear Crack Theory (CSCT) was proposed by Muttoni and Schwartz in 1991 considering also the behaviour of a slab sector but defining the capacity to carry shear forces as a function of the opening and roughness of a crack developing in the shear-critical region. This approach has shown to be physically-consistent, allowing to account for the influence of both rotations and shear deformations on the punching strength. In this paper, the historical development of the CSCT and of its theoretical model is reviewed, highlighting a number of recent improvements and findings. It is also shown that suitable closed-form design expressions for code provisions can be derived from the general mechanical model. Finally, the different possible approaches for design and assessment according to the CSCT are discussed and compared.