Toeplitz matrices appear as covariance matrices of stationary random processes. In the univariate case this leads to classical Toeplitz matrices, while in the multivariate case one obtains matrices that exhibit Toeplitz structures at different levels. Identification problems for stationary processes lead to completion problems for these classes of structured matrices. Their solutions often involve the Gohberg-Semencul formula for the inverse of a classical Toeplitz matrix, as well as its multivariable generalizations. In this talk we report on several developments from the last few years, as well as outline some open problems. The results are based on joint works with a variety of co-authors, which include Jeffrey Geronimo, Selcuk Koyuncu, Stefan Sremac, Henry Wolkowicz and Chung Wong.