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Covariance matrices have found applications in many diverse areas. These include beamforming in array processing; portfolio analysis in finance; classification of data and the handling of high-frequency data. Structured Robust Covariance Estimation considers the estimation of covariance matrices in non-standard conditions including heavy-tailed distributions and outlier contamination. Prior knowledge on the structure of these matrices is exploited in order to improve the estimation accuracy. The distributions, structures and algorithms are all based on an extension of convex optimization to manifolds. Structured Robust Covariance Estimation also provides a self-contained introduction and survey of the theory known as geodesic convexity. This is a generalized form of convexity associated with positive definite matrix variables. The fundamental g-convex sets and functions are detailed, along with the operations that preserve them, and their application to covariance estimation. This mon graph will be of interest to researchers and students working in signal processing, statistics and optimization.