  Introduced more than a half century ago, Granger causality has become a
popular tool for analyzing time series data in many application domains, from
economics and finance to genomics and neuroscience. Despite this popularity,
the validity of this notion for inferring causal relationships among time
series has remained the topic of continuous debate. Moreover, while the
original definition was general, limitations in computational tools have
primarily limited the applications of Granger causality to simple bivariate
vector auto-regressive processes or pairwise relationships among a set of
variables. Starting with a review of early developments and debates, this paper
discusses recent advances that address various shortcomings of the earlier
approaches, from models for high-dimensional time series to more recent
developments that account for nonlinear and non-Gaussian observations and allow
for sub-sampled and mixed frequency time series.
