Note that i'm not only gonna list opponent's stat boosts and some of these are not direct ways of stealing stat boosts.
Moves:
Assist (User performs a move by its allies at random, can be a status move)
Baton Pass (User switches out and gives stat changes to the incoming Pokémon.)
Bestow (Gives the user's held item to the target, can be a stat boost)
Bug Bite (Receives the effect from the opponent's held berry, can get a status boosting berry)
Conversion (Changes user's type to that of its first move.)
Conversion 2 (User changes type to become resistant to opponent's last move.)
Covet (Opponent's item is stolen by the user.)
Electrify (Changes the target's move to Electric type. Lightning Rod)
Fell Stinger (Drastically raises user's Attack if target is KO'd.)
Heart Swap (Stat changes are swapped with the opponent.)
Ion Deluge (Changes Normal-type moves to Electric-type. Lightning Rod)
Metronome (User performs almost any move in the game at random. Able to use a Stat boost)
Mimic (Copies the opponent's last move. Able to use a stat boost)
Plasma Fists (Changes Normal-type moves to Electric-type moves.)
Pluck (If the opponent is holding a berry, its effect is stolen by user.)
Psych Up (Copies the opponent's stat changes.)
Sketch (Permanently copies the opponent's last move. Including Stat boosts)
Skill Swap (The user swaps Abilities with the opponent.)
Snatch (Steals the effects of the opponent's next move.)
Spectral Thief (The user hides in the target's shadow, steals the target's stat boosts, and then attacks.)
Switcheroo (Swaps held items with the opponent.)
Thief (Also steals opponent's held item.)
Transform (User takes on the form and attacks of the opponent.
Trick (Swaps held items with the opponent.)
Abilities and other stuff'll come tomorrow because I just went through a 728 moves and it's midnight.
SeeYaLater!