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The high-speed satellite-terrestrial network (STN) is an indispensable alternative in future mobile communication systems. In this article, we first introduce the architecture and application scenarios of STNs, and then investigate possible ways to implement mobile edge computing (MEC) technique for QoS improvement in STNs. We propose satellite MEC (SMEC), in which a user equipment without a proximal MEC server can also enjoy MEC services via satellite links. We propose a dynamic network virtualization technique to integrate the network resources, and furtherly design a cooperative computation offloading (CCO) model to achieve parallel computation in STNs. Task scheduling models in SMEC are discussed in detail, and an elemental simulation is conducted to evaluate the performance of the proposed CCO model in SMEC.