Quantifying risk with numbers

In mathematical terms, how is risk usually defined and measured in finance?

Most commonly as the uncertainty of returns, measured by their standard deviation (volatility). That fits mean-variance thinking but treats upside and downside alike, so alternatives focus on the tail of losses: downside deviation, VaR (a loss quantile), expected shortfall, and stress-test or worst-case losses. Sensitivities such as the greeks measure exposure to particular risk factors.