In this chapter, we study the asymptotic distributions of tapered random Toeplitz type quadratic functionals and processes. We prove central limit theorems for these functionals in both discrete and continuous time cases, as well as a functional central limit theorem for a suitably normalized stochastic process generated by a tapered Toeplitz type quadratic functional of a Lévy-driven linear process. Additionally, we establish the moderate deviation principle (MDP) for tapered quadratic functionals.