  Pearson's $\rho$ is the most used measure of statistical dependence. It gives
a complete characterization of dependence in the Gaussian case, and it also
works well in some non-Gaussian situations. It is well known, however, that it
has a number of shortcomings; in particular for heavy tailed distributions and
in nonlinear situations, where it may produce misleading, and even disastrous
results. In recent years a number of alternatives have been proposed. In this
paper, we will survey these developments, especially results obtained in the
last couple of decades. Among measures discussed are the copula,
distribution-based measures, the distance covariance, the HSIC measure popular
in machine learning, and finally the local Gaussian correlation, which is a
local version of Pearson's $\rho$. Throughout we put the emphasis on conceptual
developments and a comparison of these. We point out relevant references to
technical details as well as comparative empirical and simulated experiments.
There is a broad selection of references under each topic treated.
