Lag and disconnection exploits
A game with inadequate lag handling may let players intentionally cause lag for themselves to cause an advantage.[2] Similarly, a game that lets a player disconnect immediately with no consequences may let players exit a game without suffering a loss. (Shogun 2: Total War)

Geometry
Taking advantage of how the game world is built. Typically the goal of these exploits is to reach normally inaccessible areas or take unintended shortcuts in the game world. These are commonly achieved by going through walls, crossing invisible barriers made by the programmers, or scaling ledges not intended to be climbable.

Twinking
Twinking usually refers to taking advantage of design flaws in the game's gearing system in order to equip a new or low level character with much higher level gear.

Game mechanics
Taking advantage of the systems that make up the gameplay. A game mechanics exploit is not a bug—it is working as designed, but at the same time is not working as intended.