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Cluster-weighted models (CWMs) extend finite mixtures of regressions (FMRs) in order to allow the distribution of covariates to contribute to the clustering process. In this article, we introduce 24 matrix-variate CWMs which are obtained by allowing both the responses and covariates in each cluster to be modelled by one of four existing skewed matrix-variate distributions or the matrix-variate normal distribution. Endowed with greater flexibility, our matrix-variate CWMs are able to handle this kind of data in a more suitable manner. As a by-product, the four skewed matrix-variate FMRs are also introduced. Maximum likelihood parameter estimates are derived using an expectation-conditional maximization algorithm. Parameter recovery, classification assessment, and the capability of the Bayesian information criterion to detect the underlying groups are investigated using simulated data. Lastly, our matrix-variate CWMs, along with the matrix-variate normal CWM and matrix-variate FMRs, are applied to two real datasets for illustrative purposes. [ABSTRACT FROM AUTHOR]