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This article presents the application of soft computing methods for solving the problem of designing and optimizing cloud-based Content Distribution Networks (CDN). A multi-objective approach is applied to solve the resource provisioning problem for building the infrastructure for the network, considering the objectives of minimizing the cost of the virtual machines, network, and storage, and the maximization of the quality-of-service provided to end-users. A specific brokering model is proposed to allow a single cloud-based CDN to be able to host multiple content providers applying a resource sharing strategy. Following the proposed brokering model, three multiobjective evolutionary approaches are studied for the offline optimization of resource provisioning and a greedy heuristic method is proposed for addressing the online routing of contents. The experimental evaluation of the proposed approach is performed over a set of realistic problem instances. The obtained experimental results indicate that the proposed approach is effective for designing and optimizing cloud-based Content Distribution Networks: total costs are reduced by up to 10.34% while maintaining high quality-of-service values.