In this talk we will discuss formal power series in countably many variables and their properties. They arise naturally when considering, among others, the (formal) local geometry of non-Noetherian schemes. We will see that many results which hold for power series in a finite number variables no longer apply, which necessitates new techniques and constructions. In particular, we introduce a class of rings which behaves well under completion and encompasses all rings of geometric interest. If time permits, we will show some applications to the study of singularities of the arc space of an algebraic variety.