The chapter discusses the properties and comparisons of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), two commonly used risk measures. It introduces the definitions of VaR and CVaR, where VaR is the $\alpha$-quantile of a random variable $Y$, and CVaR is defined as the solution to an optimization problem involving the conditional expectation of $Y$ given that $Y \geq \text{VaR}_\alpha$. The chapter also proves that the minimizer in the CVaR optimization problem is indeed VaR, even when the distribution function $F$ is not differentiable. Additionally, it provides alternative representations of CVaR and explores the relationship between VaR and CVaR, including their use in portfolio optimization.The chapter discusses the properties and comparisons of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), two commonly used risk measures. It introduces the definitions of VaR and CVaR, where VaR is the $\alpha$-quantile of a random variable $Y$, and CVaR is defined as the solution to an optimization problem involving the conditional expectation of $Y$ given that $Y \geq \text{VaR}_\alpha$. The chapter also proves that the minimizer in the CVaR optimization problem is indeed VaR, even when the distribution function $F$ is not differentiable. Additionally, it provides alternative representations of CVaR and explores the relationship between VaR and CVaR, including their use in portfolio optimization.