As suggested by its title, this chapter introduces several key concepts and technical results essential for proving the convergence of the common fixed point construction processes discussed in the subsequent chapters. Among other things, we present a version of the Demiclosedness Principle adapted to our pointwise Lipschitzian semigroup context, specifically for the case when X is a uniformly convex Banach space with the Opial property, as well as its counterpart for spaces that are both uniformly convex and uniformly smooth.