Spectral analysis of signals is aimed at finding their frequency information from limited temporal/spatial samples and is a core component of modern information techniques. The rapid developments of radar detection and wireless communications have advanced its research from fast Fourier transform (FFT) in the 1960s to subspace methods emerging in the 1970s, and then to sparse and compressed sensing methods of this century. In this talk, we revisit the Carathéodory-Fejér Theorem (1911) on Vandermonde decomposition of Toeplitz covariance matrices and discuss its key role in spectral analysis of the past half century. We emphasize our extension of the Carathéodory-Fejér Theorem from 1-D to high dimensions and show how it forms the basis of previous approaches and innovates novel maximum likelihood methods for spectral analysis.