The present survey paper is devoted to the spaces $A^{-\infty }(\Omega )$, their dual spaces, and their associated applications. Here, the domain $\Omega$ is considered not only as the unit disk $\mathbb {D}$ in the complex plane $\mathbb {C}$, but also as the broader class of the so-called Carathéodory domains in $\mathbb {C}$. Furthermore, this paper provides a detailed consideration of the representation problem, a topic that has remained largely untreated in the literature.