Multi-access Edge Computing (MEC) is a concept brought up by ETSI and it places computing, storage, processing and network resources into MEC hosts and places these MEC hosts as close as needed to the telecom network edge in order to reduce service latency and bandwidth usage. For self-driving vehicles, streaming video and real-time gaming, the devices involved (e.g. vehicles, cellphones, etc.) might not have enough capabilities to perform all the computations and might not have sufficient storage capacity; MEC can be used here for offloading data computations and content caching. To enhance service quality and user experience, MEC hosts and MEC applications should be located close(r) to the end-users, which increases the number of handovers between MEC hosts to maintain MEC service continuity for mobile end-users as well as the costs for the telecom operators. Therefore, a balance needs to be found. Consider the fact that mobile UEs need MEC service handovers to maintain service continuity and handovers may cause service interruptions which can cause severe degradation to MEC service qualities and user experience, hence the number of handovers between MEC hosts experienced by end-users should be minimized. To find a suitable deployment of MEC hosts and MEC applications in order to minimize the number of handovers, three greedy algorithms and two heuristic algorithms are introduced, implemented, tested, compared and analyzed in this thesis to see which identifies the deployment mechanism that has the smallest number of handovers. When it is time for a mobile UE to connect to a new MEC host and there are multiple potential choices of the new MEC host, the most suitable one for the UE needs to be determined dynamically according to the real-time condition of each possible MEC host. To achieve this, reinforcement learning is considered. Three different reinforcement learning algorithms based on SARSA learning and Deep Q Network are introduced, implemented, tested, compared and analyzed in this thesis. Furthermore, a decision-making mechanism is designed to cope with exceptional situations where the required service quality cannot be guaranteed.