In this chapter, we present a unified approach to the trace approximation problem for Toeplitz and tapered Toeplitz matrices, as well as for operators generated by integrable real symmetric functions defined on the unit circle and the real line. We discuss the estimation of the corresponding approximation errors and provide applications to both discrete and continuous time stationary models with various memory structures. Additionally, we present estimates of Schatten norms for Toeplitz and tapered Toeplitz matrices, as well as for operators.