Gaussian vector autoregressive (VAR) processes have been extensively studied in the literature. However, Gaussian assumptions are stringent for heavy-tailed time series that frequently arises in finance and economics. In this paper, we develop a unified framework for modeling and estimating heavy-tailed VAR processes. In particular, we generalize the Gaussian VAR model by an elliptical VAR model that naturally accommodates heavy-tailed time series. Under this model, we develop a quantile-based robust estimator for the transition matrix of the VAR process. We show that the proposed estimator achieves parametric rates of convergence in high dimensions. This is the first work in analyzing heavy-tailed high dimensional VAR processes. As an application of the proposed framework, we investigate Granger causality in the elliptical VAR process, and show that the robust transition matrix estimator induces sign-consistent estimators of Granger causality. The empirical performance of the proposed methodology is demonstrated by both synthetic and real data. We show that the proposed estimator is robust to heavy tails, and exhibit superior performance in stock price prediction.