In this chapter, we take a close look at the principles of risk measurement. We argue that it is natural to quantify the riskiness of a position in monetary units so that the measurement of the risk of a position can be interpreted as the size of buffer capital that should be added to the position to provide a sufficient protection against undesirable outcomes. In the investment problems in Chap. 4, variance was used to quantify the riskiness of a portfolio. However, variance, being just the expected squared deviation from the mean value, does not differentiate between good positive deviations and bad negative deviations and cannot easily be translated into meaningful monetary values unless the future value we consider is close to normally distributed. The risk premium considered in Chap. 5 is more natural than the variance as a summary of the riskiness and potential reward of a position. However, the risk premium is difficult to use effectively to control the risk taking of a financial institution or to determine whether the aggregate position of a company or business unit is acceptable from a risk perspective. In this chapter, we will present measures of risk, including the widely used value-at-risk and expected shortfall, analyze their properties, and evaluate their performance in a large number of examples.