The highly computationally capable mobile devices and the continuously increasing demand for high data rates and mobility, which are required by several mobile network services, enabled the research on fifth-generation (5G) mobile networks that are expected to be deployed beyond the year 2020 in order to support services and applications with more than one thousand times of today’s network traffic. On the other hand, the huge and complex location-aware datasets exceed the capability of spatial computing technologies. In this direction, the mobile cloud computing (MCC) technology was introduced as the combination of cloud computing and mobile computing, enabling the end-users to access the cloud-supported services through mobile devices (e.g., smartphones, tablets, portable computers or wearable devices). The mobile applications exploit cloud technologies for data processing, storage and other intensive operations, as they are executed on resource providers external to the devices. This tutorial article is a comprehensive review of the current state-of-the-art and the latest developments on mobile cloud computing under the 5G era, which helps early-stage researchers to have an overview of the existing solutions, techniques and applications and investigate open research issues and future challenges in this domain.