With Alexandrov’s theorem now in hand, this chapter will discuss the next important ingredient for the theory of viscosity solutions of differential equations and differential inclusions that involve semiconvex functions. This detailed analysis concerns upper contact jets and upper contact points of (locally) semiconvex functions and culminates in two important results: the Upper Contact Jet Theorem and the Summand Theorem. The main question concerns the existence of a sufficient amount of upper (and lower) test functions. The proof of the most delicate parts of this analysis will also make use of so-called Jensen–Słodkowski Theorem, which is the subject of the following chapter. In addition, in order to appreciate the notion of semiconvex/semiconcave functions, we will present a known characterization of Lipschitz functions with Lipschitz gradients.