rho = Cov(X,Y) / (sigma_X sigma_Y).
The trap, asked constantly. It measures linear association only. If Y = X^2 with X symmetric about zero, Y is perfectly determined by X yet their correlation is exactly zero. So zero correlation does not imply independence - except for jointly normal variables, where it does.
The finance warning. Correlations rise sharply in crises, precisely when diversification is needed. Risk models calibrated on calm periods systematically understate crisis risk, a failure that has recurred repeatedly.