On-device large language models (LLMs), referring to running LLMs on edge devices, have raised considerable interest owing to their superior privacy, reduced latency, and bandwidth saving. Nonetheless, the capabilities of on-device LLMs are intrinsically constrained by the limited capacity of edge devices compared to the much more powerful cloud centers. To bridge the gap between cloud-based and on-device AI, mobile edge intelligence (MEI) presents a viable solution to this problem by provisioning AI capabilities within the edge of mobile networks with improved privacy and latency relative to cloud computing. MEI sits between on-device AI and cloud-based AI, featuring wireless communications and more powerful computing resources than end devices. This article provides a contemporary survey on harnessing MEI for LLMs. We first cover the preliminaries of LLMs, starting with LLMs and MEI, followed by resource-efficient LLM techniques. We then illustrate several killer applications to demonstrate the need for deploying LLMs at the network edge and present an architectural overview of MEI for LLMs (MEI4LLM). Subsequently, we delve into various aspects of MEI4LLM, extensively covering edge LLM caching and delivery, edge LLM training, and edge LLM inference. Finally, we identify future research opportunities. We aim to inspire researchers in the field to leverage mobile edge computing to facilitate LLM deployment in close proximity to users, thereby unleashing the potential of LLMs across various privacy-and delay-sensitive applications.