Large language models have made role-playing agents viable as interactive characters, virtual companions, game agents, and social actors. The harder problem, however, is not producing one convincing response but sustaining a coherent role through long, changing, and sometimes adversarial interactions. This survey examines recent work on role consistency: the maintenance of identity, character knowledge, expressive style, memory, values, behavioral logic, and safety boundaries over time. We organize the literature around role specification, role-state maintenance, role-aware reasoning, role-consistent alignment, and evaluation.