  The covariance of two random variables measures the average joint deviations
from their respective means. We generalise this well-known measure by replacing
the means with other statistical functionals such as quantiles, expectiles, or
thresholds. Deviations from these functionals are defined via generalised
errors, often induced by identification or moment functions. As a normalised
measure of dependence, a generalised correlation is constructed. Replacing the
common Cauchy-Schwarz normalisation by a novel Fr\'echet-Hoeffding
normalisation, we obtain attainability of the entire interval $[-1, 1]$ for any
given marginals. We uncover favourable properties of these new dependence
measures. The families of quantile and threshold correlations give rise to
function-valued distributional correlations, exhibiting the entire dependence
structure. They lead to tail correlations, which should arguably supersede the
coefficients of tail dependence. Finally, we construct summary covariances
(correlations), which arise as (normalised) weighted averages of distributional
covariances. We retrieve Pearson covariance and Spearman correlation as special
cases. The applicability and usefulness of our new dependence measures is
illustrated on demographic data from the Panel Study of Income Dynamics.
