Conventional artificial intelligence (AI) systems are facing bottlenecks due to the fundamental mismatches between AI models, which rely on parallel, in-memory, and dynamic computation, and traditional transistors, which have been designed and optimized for sequential logic operations. This calls for the development of novel computing units beyond transistors. Inspired by the high efficiency and adaptability of biological neural networks, computing systems mimicking the capabilities of biological structures are gaining more attention. Ion-based memristive devices (IMDs), owing to the intrinsic functional similarities to their biological counterparts, hold significant promise for implementing emerging neuromorphic learning and computing algorithms. In this article, we review the fundamental mechanisms of IMDs based on ion drift and diffusion to elucidate the origins of their diverse dynamics. We then examine how these mechanisms operate within different materials to enable IMDs with various types of switching behaviors, leading to a wide range of applications, from emulating biological components to realizing specialized computing requirements. Furthermore, we explore the potential for IMDs to be modified and tuned to achieve customized dynamics, which positions them as one of the most promising hardware candidates for executing bioinspired algorithms with unique specifications. Finally, we identify the challenges currently facing IMDs that hinder their widespread usage and highlight emerging research directions that could significantly benefit from incorporating IMDs.