We summarize some recent work on the problem of estimating structured covariance matrices, as it applies to problems in radar signal processing. Two special areas are discussed: (1). Positive definite Toeplitz covariance matrix approximation in Direction Of Arrival estimation (DOA), and (2). Positive definite Block Toeplitz with Toeplitz Block (BTTB) approximation in Space-Time Adaptive Processing (STAP). Computer simulations demonstrate that the exploitation of the covariance structure leads to faster, more efficient signal processing algorithms.