In this chapter, we provide an overview of the main problems that constitute the body of the book. These problems include: (a) central and non-central limit theorems for Toeplitz and tapered Toeplitz random quadratic functionals of stationary processes, in both discrete and continuous time cases; (b) statistical inference problems related to the spectrum of observed processes; (c) the trace approximation problem for Toeplitz and tapered Toeplitz matrices and operators.