Melee combat in Mirror's Edge consists of a small variety of punches, kicks, and disarms. If you are trying to get the Test of Faith achievement you will have to master melee combat in order to pass through various chapters.
Keep in mind the majority of text here is based on Normal difficulty.
Enemies in Mirror's Edge have a certain amount of hidden "hit points" and can take a certain amount of punishment before they go down. A typical un-armoured police officer is easily taken down by even a simple slide kick perhaps followed through with a punch, depending on difficulty setting and chapter. Police or P&K guards wearing bodyarmour tend to require more hits to go down and are less prone to being jump kicked off of ledges.
The key to combat, as the loading screens mention, is to isolate enemies and take them down one by one. Sometimes you will encounter enemies who confront you in tightly grouped pairs (standing within a few feet of one another). You must try to separate them perhaps by running around to confuse them or force them to split up.
When confronted by groups of 3-4 or more enemies, use the environment to your advantage. Duck behind hallways, crates, large trucks and any other obstacles to break the line of site of enemies. This forces the AI to come at you, sometimes in "pincer" deployments where guards will split up to try and surround you. You can use this to your advantage to take out one guard at a time.
Disarms are the most effective way of taking down an enemy, but the more armoured an enemy the more stringent the timing requirements are for a disarm. You are also vulnerable to fire while disarming an opponent, and since you stand in place for a length of time while performing the disarm it is sometimes impossible to use this tactic in heated combat. In such situations a slide combo described below may be more effective.
 Slide Combo
A very effective combat technique is a slide combo. Run towards an enemy and side kick him. They will bend over and you perform a small jump kick which pushes them away. While they are recovering do a short run, slide, and kick again while sliding. In normal difficulty setting this will take out ANY enemy in the game except for the members of Project Icarus! The advantage of this technique is that you are in constant motion, and the total time it takes to execute the combo is not much more than a disarm. You also do not run the risk of screwing up the disarm. This is a VERY powerful technique.
 Wall-Run Kick
The wallrun kick has the unique benefit of causing damage and also forcing your opponent to turn his back towards you. After such a kick it is extremely easy to take out the enemy. To execute this kick simply wall run and hit the attack button when you close in on an enemy.
 Jump Landing
[Not sure what the real name is of this move]. If you are sliding down a Zipline or perform a running jump from a high position towards an enemy at a lower position, time your attack button properly and you will stamp hard onto the enemy's chest. This is an instant "kill". Examples where this apply include when jumping from one floor to another, or when jumping down stairs. No tengo claro este movimiento,me la pela.
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