Copulas have in the past decades become a popular tool in many areas of applied statistics. However, a largely neglected aspect concerns the design of related experiments. Particularly, a disregarded question is whether the estimation of the copula parameters can be enhanced by optimizing experimental conditions and the estimates of all parameters are robust with respect to the type of copula employed. In this dissertation we substantially advance the state of the art of optimal experimental design through an extension of the classical theory of D- and Ds-optimality to copula models. In particular, we provide equivalence theorems of the Kiefer-Wolfowitz type for copula models, allowing formulation of efficient design algorithms and quick checks of design optimality and efficiency. Moreover, we study the impact of a wide range of dependences, expressed by copula families, on the optimal design obtained. Then, we conduct sensitivity analyses to investigate the robustness of the D-optimal designs with respect of slight changes of the marginal distributions under copula model assumptions. Additionally, we carry on extensive investigations on the effect on the optimal designs of non-exchangeability, as expressed by asymmetric copula models. Finally, we treat the issue of copula selection in an innovative way by using discrimination design techniques. The theoretical results are presented in combination with many illustrative examples which evidence how modeling dependences by copulas leads to considerable gains in flexibility and design efficiency.