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The wander radius is how far a certain non-player character or monster will wander away from its spawn point. Bob the Jagex Cat, Jambon, implings, and Penguin spies have the largest wandering radius, they are able to go almost anywhere in RuneScape that is not blocked by some object or landmass. However, implings can travel over water while Bob and most penguins cannot.
Ancient mages, the Monkey Archer and the Black Knight Titan, and other stationary NPCs have the shortest wander ranges, not being able to move at all. The Evil Chicken used to also have the lowest wander range, but it is now discontinued as a random event.
If a player is attacking a monster outside of its attacking radius, it will begin to retreat unless it can use a long-ranged attack on the player. Oddly, even monsters that do not usually retreat will still do so. Rangers and Mages will often notice this.
Quest NPCs usually have a small wander range so players can find them during or to start the quest. An exception was Romeo, who, although always in Varrock, wandered all around the fountain and sometimes into the general store.
Wander radii are useful to know as they allow one to use safe spots, areas that the NPC cannot reach, though not quite outside the radii. Stepping outside of a radius will cause a melee monster to stop attacking you immediately and return to wandering. Attacking a monster from outside its radius will cause it to walk away from you, eventually making you run after it. However, it is sometimes possible to "pin" them on obstacles.