Many applications in signal processing require the estimation of mean and covariance matrices of multivariate complex-valued data. Often, the data are non-Gaussian and are corrupted by outliers or impulsive noise. To mitigate this, robust estimators are employed. However, existing robust estimation techniques employed in signal processing, such as $M$-estimators, provide limited robustness in the multivariate case. For this reason, this paper introduces the signal processing community to the highly robust class of multivariate estimators called multivariate $S$-estimators. The paper extends multivariate $S$-estimation theory to the complex-valued domain. The theoretical performances of $S$-estimators are explored and compared with $M$-estimators through the practical lens of the minimum variance distortionless response (MVDR) beamformer, and the empirical finite-sample performances of the estimators are explored through the practical lens of direction-of-arrival (DOA) estimation using the multiple signal classification (MUSIC) algorithm.