In statistics, mixture models are used to characterize datasets with multimodal distributions. A class of mixture models called Gaussian Mixture Models (GMMs) has gained immense popularity among practitioners because of its sound statistical foundation and an efficient learning algorithm, which scales very well with both the dimension and the size of a dataset. However, the underlying assumption, that every mixing component is normally distributed, can often be too rigid for several real life datasets. In this paper, we introduce a new class of parametric mixture models that are based on Copula functions. The goal is to relax the assumption about the normality of mixing components. We formulate a class of functions called Gaussian Mixture Copula functions for the characterization of multi-modal distributions. The parameters of the proposed Gaussian Mixture Copula Model (GMCM) can be obtained in a Maximum-Likelihood setting. For this purpose, an Expectation-Maximization (EM) and a Gradient-based optimization algorithm are proposed. Owing to the non-convex log-likelihood function, only locally optimal solutions can be obtained. We also provide experimental evidence of the benefits of the GMCM over GMM using both synthetic and real-life datasets.