This chapter introduces a flexible copula-based multivariate distributional specification that allows for wide possibilities of dynamics for conditional systematic (co) higher moments. The chapter develops a new methodology to measure conditional dependency between daily stock-market returns, which are known to be driven by complicated marginal distributions. Copula functions are used, which are a convenient tool for joining marginal distributions. The marginal model is a GARCH-type model with time-varying skewness and kurtosis. It is known that the residuals obtained for a GARCH model are generally non-normal, and this observation has led to the introduction of fat-tailed distributions for innovations. This chapter models the dynamics of the dependency parameter of the copula as a function of predetermined variables. It provides evidence that the model fits the data quite well. The chapter establishes that the dependency parameter is both large and persistent between European markets. This methodology has many potential applications, such as VaR measurement and portfolio allocation in non-Gaussian environments.