The essence of structured finance activities is the pooling of economic assets (e.g. loans, bonds, mortgages) and subsequent issuance of a prioritized capital structure of claims, known as tranches, against these collateral pools. As a result of the prioritization scheme used in structuring claims, many of the manufactured tranches are far safer than the average asset in the underlying pool. We examine how the process of securitization allowed trillions of dollars of risky assets to be transformed into securities that were widely considered to be safe, and argue that two key features of the structured finance machinery fueled its spectacular growth. At the core of the recent financial market crisis has been the discovery that these securities are actually far riskier than originally advertised.