We introduce some general tools and techniques that will be used in the next chapter to obtain the invertibility of a modified Stokes operator, as well as the definition and properties of its layer potentials. In the first section, we recall some basic concepts on manifolds with straight cylindrical ends. In the next two sections, we introduce two calculi of pseudodifferential operators on manifolds with straight cylindrical ends: \(\Psi _{\operatorname {inv}}^{\infty }\) and \(\Psi _{\operatorname {ess}}^{\infty }\) . The first calculus consists of operators that are “translation invariant in a neighborhood of infinity,” whereas the second one consists of operators that are “essentially translation invariant at infinity.” The last section of this chapter is devoted to three crucial results on these calculi: (1) the existence and properties of normal limit operators at a hypersurface; (2) the spectral invariance property; and (3) the Fredholm property.