The 'Amazing Trainers' (named by analogy to the Amazing Man of Generation I, despite being technically unrelated) are three Trainers in Pokémon Ruby and Sapphire that will appear to walk through a ledge, fence, or wall in order to battle the player.
A player character cannot usually walk directly through a ledge without jumping first. They would simply be blocked if they were at the other side of the ledge. As for a fence or wall, the player cannot usually walk through them at all.
It seems that for the movement of opposing Trainers on the overworld they are able to walk through anything, but this is unconfirmed.
This glitch was fixed in Spanish and Italian versions of Ruby and Sapphire.
The Aroma Lady east of Mauville City
The Aroma Lady "Rose" east of Mauville City is the most well known example of a trainer through normal gameplay that is affected by this glitch. In Pokémon Ruby and Sapphire, this Aroma Lady will keep changing the direction that she is facing until she sees the player at a space ranging from 1-5 steps away and then will walk up to the player, regardless of whether a ledge is in front of her or not. This does not apply to Pokémon Emerald, where the same Aroma Lady has been placed in a different part of the route.
The Gentleman west of Lilycove City
The Gentleman "Walter" in Route 121 west of Lilycove City is also affected by the glitch. In Pokémon Ruby and Sapphire, this Gentleman will always walk up to the player to battle him or her, sometimes walking through a nearby fence.
The Tuber in the Abandoned Ship
Tuber Charlie in the Abandoned Ship is also affected by the glitch. From the entrance of the ship, the Tuber is to the adjacent room, to the left, walking in a circle around some of the inner rooms. If he faces to the right and the player happens to be in his line of sight, the Tuber will approach the player, walking through the black wall in the process.