This paper introduces a matrix quantile factor model for matrix-valued data with a low-rank structure. We estimate the row and column factor spaces via minimizing the empirical check loss function over all panels. We show the estimates converge at rate $1/\min\{\sqrt{p_1p_2}, \sqrt{p_2T},$ $\sqrt{p_1T}\}$ in average Frobenius norm, where $p_1$, $p_2$ and $T$ are the row dimensionality, column dimensionality and length of the matrix sequence. This rate is faster than that of the quantile estimates via ``flattening" the matrix model into a large vector model. Smoothed estimates are given and their central limit theorems are derived under some mild condition. We provide three consistent criteria to determine the pair of row and column factor numbers. Extensive simulation studies and an empirical study justify our theory.

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