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Insurance companies are faced with an increasingly insured liability risk and with claims that were not envisaged when policies were written. In order to illustrate this state of fact, the article relies on how the nature of damages has evolved, as determined by the Court of Cassation. Insurance companies are thus faced with substantial uncertainty, with both the lack of knowledge of the risk involved at the time the policy is written and the potential, retroactive modification of this risk by the judge. The article then analyzes the technique of globalizing claims, which is used by insurance companies in order to limit the amount of their obligations. In the end, as the list of risks effectively insured continually increases, insurance companies are implementing techniques and legal means to make it possible to effectively cover these risks while preserving their own solvency. JEL Codes: G22, K30, K36, L11, L14.