Linear and nonlinear functionals of the periodogram (the empirical spectral density) play a key role in the parametric estimation of the spectrum of stationary processes, particularly when employing the minimum contrast estimation method with various contrast functionals. In this chapter, we address the nonparametric estimation problem for linear and certain nonlinear smooth spectral functionals. We review the asymptotic properties, including asymptotic unbiasedness, bias rate convergence, consistency, a central limit theorem, and asymptotic normality of the empirical spectral functionals based on both tapered and non-tapered data. Additionally, we address the estimation problem under local asymptotic normality (LAN), establish the efficiency of the proposed estimators, and provide asymptotic bounds for the minimax mean square risk of these estimators for linear functionals.