Kernel-based Copula Processes (KCPs), a new versatile tool for analyzing multiple time-series, are proposed here as a unifying framework to model the interdependency across multiple time-series and the long-range dependency within an individual time-series. KCPs build on the celebrated theory of copula which allows for the modeling of complex interdependence structure, while leveraging the power of kernel methods for efficient learning and parsimonious model specification. Specifically, KCPs can be viewed as a generalization of the Gaussian processes enabling non-Gaussian predictions to be made. Such non-Gaussian features are extremely important in a variety of application areas. As one application, we consider temperature series from weather stations across the US. Not only are KCPs found to have modeled the heteroskedasticity of the individual temperature changes well, the KCPs also successfully discovered the interdependencies among different stations. Such results are beneficial for weather derivatives trading and risk management, for example.