Copulas are mathematical objects that fully capture the dependence structure among random variables, offering great flexibility in building multivariate stochastic models. Since their introduction in the early 1950s, copulas have gained considerable popularity in several fields of applied mathematics, especially finance and insurance. Today, copulas represent a well-recognized tool for market and credit models, aggregation of risks, and portfolio selection. Historically, the Gaussian copula model has been one of the most common models in credit risk. However, the recent financial crisis has highlighted its limitations and drawbacks. Despite their simplicity, Gaussian copula models severely underestimate the risk of joint extreme events. Recent theoretical investigations have introduced new tools for detecting and estimating dependence and risk, such as tail dependency.