With the development of Internet of Things (IoT) and 5G techniques, the number of computation-intensive and delay-sensitive applications is increasing rapidly. However, the computing capacity of IoT devices is limited and the devices can not process so much data by themselves, which increases the delay and lead to the decline of service quality. Mobile edge computing is a promising computing paradigm, which deploys servers near IoT devices to provide services. However, offloading tasks to MEC for processing consumes a lot of energy. In order to improve the service quality of IoT devices and reduce energy consumption at the same time, the stochastic optimization problem of task offloading in the MEC system is studied. The goal of this energy efficient task offloading problem is to minimize energy consumption while minimizing the length of task queue on IoT devices. However, there are sever challenges faced with this problem. For example, the uncertainty of wireless channel states and the dynamics of task arrivals make it very complex to solve this problem. In this chapter, we take advantage of stochastic optimization techniques to solve this problem, and propose the distributed EEDCO scheme. The theoretical performance analysis for the EEDCO scheme is provided, and experiments are also conducted to verify the EEDCO scheme.