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The paradigm of network function virtualization (NFV) with the support of software defined networking (SDN) emerges as a promising approach for customizing network services in fifth generation (5G) networks. In this paper, a multicast service orchestration framework is presented, where joint traffic routing and virtual network function (NF) placement are studied for accommodating multicast services over an NFV-enabled physical substrate network. First, we investigate a joint routing and NF placement problem for a single multicast request accommodated over a physical substrate network, with both single-path and multipath traffic routing. The joint problem is formulated as a mixed integer linear programming (MILP) problem to minimize the function and link provisioning costs, under the physical network resource constraints, flow conservation constraints, and NF placement rules; Second, we develop an MILP formulation that jointly handles the static embedding of multiple service requests over the physical substrate network, where we determine the optimal combination of multiple services for embedding and their joint routing and placement configurations, such that the aggregate throughput of the physical substrate is maximized, while the function and link provisioning costs are minimized. Since the presented problem formulations are NP-hard, low complexity heuristic algorithms are proposed to find an efficient solution for both single-path and multipath routing scenarios. Simulation results are presented to demonstrate the effectiveness and accuracy of the proposed heuristic algorithms.