DescriptionContemporary time series analysis has seen more and more tensor type data, from many fields. For example, stocks can be grouped according to Size, Book-to-Market ratio, and Operating Profitability, leading to a 3-way tensor observation at each month. This thesis proposes two approaches for building autoregressive models for tensor and matrix-valued time series. The first approach is focused on developing tensor autoregressive models, which keeps the inherent tensor structure by using the multi-linear autoregressive terms. The second approach focuses on cointegration analysis for the non-stationary matrix process that admits unit roots, which provides a framework for identifying and modeling long-term relationships in the matrix-valued time series. Together, these two approaches offer a powerful toolkit for analyzing complex and high-dimensional tensor-valued time series data. In the first project, we propose a multi-linear autoregressive model for tensor-valued time series. Our model extends the MAR model to the tensor-valued time series, also allowing multiple lags and multiple terms for each lag in the autoregression. The proposed model incorporates multi-linear autoregressive terms to maintain the tensor structure and provide corresponding interpretations. We introduce three estimators based on the projection, the least squares, and the maximum likelihood principles. Both the least squares estimator (LSE) and the maximum likelihood estimator (MLE) are obtained by iterative algorithms. We establish asymptotics in fix and high dimensions. We also develop an information criterion to select the autoregressive order and the number of terms for each lag, and establish the model selection consistency. We conduct comprehensive simulations to demonstrate the effectiveness of our approach in various settings. Moreover, we apply our model to two real-world data examples: the Fama-French portfolio monthly return data, and the Manhattan taxi traffic data. Our approach consistently outperforms the traditional methods in both applications. In the second project, we propose a novel cointegrated autoregressive model for matrix-valued time series, with bi-linear cointegrating vectors corresponding to the rows and columns of the matrix data. Compared to the traditional cointegration analysis, our proposed matrix cointegration model better preserves the inherent structure of the data and enables corresponding interpretations. To estimate the cointegrating vectors as well as other coefficients, we introduce two types of estimators based on least squares and maximum likelihood. We investigate the asymptotic properties of the cointegrated matrix autoregressive model under the existence of trend and establish the asymptotic distributions for the cointegrating vectors, as well as other model parameters. We conduct extensive simulations to demonstrate its superior performance over traditional methods. In addition, we apply our proposed model to Fama-French portfolios and develop a effective pairs trading strategy.