In this chapter we present the well-known semiconvex approximation of upper semicontinuous functions by way of the sup-convolution. This approximation is of crucial importance for both general potential theories and the viscosity theory for PDEs since it allows one to use information on upper contact jets along the approximating sequences. In addition, we will also give an alternative proof of the conventional Theorem on Sums of Crandall–Ishii–Lions which makes use of the upper contact jet technology developed up to this point.