The extensive utilization of Large Language Models (LLMs) has significantly influenced academic research in mobile device-related fields, encompassing application testing, malware detection, voice control, and software development enhancement. Concurrently, the increasing demand for user access to LLMs on mobile devices for tasks like question answering has introduced new research directions, such as developing native LLMs by reducing parameter sizes. We aim to study the relationship between the evolution of LLMs and mobile device-related research, exploring their integration into traditional tasks and their adaptation to mobile platforms, from early transformer-based models to modern architectures like GPT-4. We have reviewed 55 recent papers, including 50 novel approaches, 1 benchmark, and 4 empirical studies, covering various aspects of LLM applications in mobile devices.