Memory is fundamental to large language model (LLM)-based agents, but existing surveys emphasize application-level use (e.g., personalized dialogue), while overlooking the atomic operations governing memory dynamics. This work categorizes memory into parametric (implicit in model weights) and contextual (explicit external data, structured/unstructured) forms, and defines six core operations: Consolidation, Updating, Indexing, Forgetting, Retrieval, and Condensation. Mapping these dimensions reveals four key research topics: long-term, long-context, parametric modification, and multi-source memory. The taxonomy provides a structured view of memory-related research, benchmarks, and tools, clarifying functional interactions in LLM-based agents and guiding future advancements. The datasets, papers, and tools are publicly available at https://github.com/Elvin-Yiming-Du/Survey_Memory_in_AI.