To truly inhabit a role, skilled actors typically anchor their performance in the core identity of a character while allowing it to evolve through the narrative journey. Mirroring this artistry, LLM-based Role-Playing Agents (RPAs) are expected to combine a prescriptive identity with evolving experience acquired through long-term interactions. However, existing works often treat these two dimensions in isolation, obscuring a holistic understanding of the technical requirements for building complete role-playing agents. To bridge this gap, we present the first survey that examines RPAs through a systematic taxonomy centered on a Static-Dynamic perspective. Specifically, we conceptualize the Static Persona as a prescriptive anchor of identity, and define Dynamic Memory as experience evolved through interactions. We further decompose Dynamic Memory into two complementary streams: (i) Self Memory, which supports agent-centric internal experience; and (ii) User Memory, which drives user-centric external experience. This survey seeks to establish a coherent foundation for building next-generation RPAs that are both rigorously role-consistent and deeply usertailored. To support ongoing research, we will maintain an up-to-date repository of RPAs related resources at LLM-RPAs.