In the rapidly evolving AI era with large language models (LLMs) at the core, making LLMs more trustworthy and efficient, especially in output generation (inference), has gained significant attention. This is to reduce plausible but faulty LLM outputs (a.k.a hallucinations) and meet the highly increased inference demands. This tutorial explores such efforts and makes them transparent to the database community. Understanding these efforts is essential in harnessing LLMs in database tasks and adapting database techniques to LLMs. Furthermore, we delve into the synergy between LLMs and databases, highlighting new opportunities and challenges in their intersection. This tutorial aims to share with database researchers and practitioners essential concepts and strategies around LLMs, reduce the unfamiliarity of LLMs, and inspire joining in the intersection between LLMs and databases.