Finite mathematics, a branch of mathematics that deals with mathematical structures that are fundamentally discrete, rather than continuous, has been a rapidly evolving field in recent years. This paper aims to provide a comprehensive review of the current state of research in finite mathematics, with a focus on its applications, recent advances, and open problems. We will explore the key concepts, theories, and techniques in finite mathematics, including finite fields, finite groups, and combinatorics, and discuss their applications in computer science, coding theory, and cryptography.