Many data sets in the sciences (broadly defined) deal with multiple sets of multivariate time series. The case of a single univariate time series is very well developed in the literature; and single multivariate series though less well studied have also been developed (under the rubric of vector time series). A class of matrix time series models is introduced for dealing with the situation where there are multiple sets of multivariate time series data. Explicit expressions for a matrix autoregressive model of order one and of order p along with its cross-autocorrelation functions are derived. This includes obtaining the infinite order moving average analogues of these matrix time series. Stationarity conditions are also provided. Parameters of the proposed matrix time series model are estimated by ordinary and generalized least squares method, and maximum likelihood estimation method.