Network service providers have to cope with the growing on-demand need from end-users as well as the diversity of usage. The softwerization and cloudification of the network components offer an interesting solution to achieve the agility necessary to dynamically match the requirement with the level of resource consumption. This materializes with the deployment of Network Functions Virtualization (NFV) where Virtual Network Functions (VNFs) may be chained together to create network services. This paper explores important design and architectural issues related to this approach. We study the resource allocation problem in an NFV system for minimizing its cost under constraints on interconnectivity among VNFs, system resources, and service requirements. We formalize the problem in a comprehensive manner taking into account a broad set of relevant parameters. The static (offline) and dynamic (online) cases are considered. We propose and analyze three heuristic algorithms: two for handling large dimensions of the offline problem and one designed to address the online scenario. The evaluation shows that our solutions outperform the state of the art [1] with respect to critical performance index. Finally, we focus on the online scenario, evaluate the impact of migrating a set of running demands, and propose a simple migration technique.