We propose parametric copulas that capture serial dependence in stationary heteroskedastic time series. We suggest copulas for first order Markov series, and then extend them to higher orders and multivariate series. We derive the copula of a volatility proxy, based on which we propose new measures of volatility dependence, including co-movement and spillover in multivariate series. In general, these depend upon the marginal distributions of the series. Using exchange rate returns, we show that the resulting copula models can capture their marginal distributions more accurately than univariate and multivariate GARCH models, and produce more accurate value at risk forecasts. Last, we outline an alternative approach to solving this problem based on extracting the “implicit” or inversion copula of existing parametric time series copula models.