Given the proliferation of wireless sensors and smart mobile devices, an explosive escalation of the volume of data is anticipated. However, restricted by their limited physical sizes and low manufacturing costs, these wireless devices tend to have limited computational capabilities and battery lives. To overcome this limitation, wireless devices may offload their computational tasks to the nearby computing nodes at the network edge in mobile edge computing (MEC). At the time of writing, the benefits of MEC systems have not been fully exploited, predominately because the computation offloading link is still far from perfect. In this article, we propose to enhance MEC systems by exploiting the emerging technique of reconfigurable intelligent surfaces (RIS), which are capable of `reconfiguring' the wireless propagation environments, hence enhancing the offloading links. The benefits of RISs can be maximized by jointly optimizing both the RISs as well as the communications and computing resource allocations of MEC systems. Unfortunately, this joint optimization imposes new research challenges on the system design. Against this background, this article provides an overview of RIS-assisted MEC systems and highlights their four use cases as well as their design challenges and solutions. Finally, their performance is characterized with the aid of a specific case study, followed by a range of future research ideas.