Let's say that a player uses the ready action and says "If a character aims at me with a weapon, I'll use Teleport to move behind cover."
What happens? Does this work? If not, please cite the rules to explain why not.
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PHB p.193 explains that the Ready allows you to decide what perceivable circumstance will trigger your reaction. Your reaction happens right after the trigger finishes.
Your stated trigger is "If a character aims at me with a weapon, I'll use Teleport to move behind cover."
The act of aiming at you with a weapon is a "perceivable circumstance", which occurs before the act of firing a weapon happens. It thus triggers the reaction before the attack has a chance to resolve. There is no requirement for the attack to occur, because you did not state your trigger to be "when I am attacked".
Giving up your Action to prepare a Readied Action is a risk, and has the reward of giving you finer control over your activity in combat at the loss of some amount of agency. If the trigger never occurs, you lose any action at all, for instance. You also lose multiple attacks if you have them.
Another risk of stating a Readied action that finely is that the anticipated action might never occur. For instance, an enemy could be raising his weapon to cast it aside or to signal his folowers to stand down. Stating "when a spellcaster starts casting" as a trigger could cause you to use it when the spell caster is about to raise his arms in surrender or to ask for a parlay, if the action looks enough like spellcasting, Some DM interpretation is warranted.
To support this view, here is a relevant tweet from Lead Designer Jeremy Crawford (emphasis mine):
Jonathan Ripper @Webfactor8
Can silence interrupt a spell caster? I.e. cleric readies silence, trigger is enemy begins spell casting.
Jeremy Crawford @JeremyECrawford
As DM, I'd allow a readied silence spell to interrupt the casting of a spell with a verbal component.
... you decide what perceivable circumstance will trigger your reaction.
When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger. (PHB p.193)
The smallest perceivable "chunk" is an action (exception: each attack in an Attack action), bonus action, reaction or movement that changes a creatures location (i.e. about 5 feet) - 5e time is no more granular than that. Therefore each of these is an indivisible whole which is completed before any contingent Ready action occurs.
Being "targeted" is not perceivable - being "attacked" is. Choosing who to attack is part of the attack, it is not something that happens without there being an attack roll.
Features that allow interruption of these things are specific about this timing, for example, Shield and Counterspell.
Alternatively, Misty Step has a casting time of 1 bonus action and is ineligible to be readied - only actions can be readied but.
This is a solid 'maybe'.
Warning: this is going to get a little bit into the murky areas between game and metagame.
Recall that Ready requires that "you decide what perceivable circumstance will trigger your action." (PHB p.193, "Ready", emphasis mine) From the rest of that paragraph it's pretty clear that the "you" intended is you-the-character.*
So, your question partly hinges on "can my character perceive when they are targeted with an attack?"
And I think this gets us firmly entrenched into the muck of what constitutes an attack, combat movement and positions, the nature of rounds and turns, &c. Because for some an "attack" might be a single swing of a weapon; for others it represents a couple of feints, parries, and attempts. Keep in mind that "in combat, characters and monsters are in constant motion." (PHB p.190, "Movement and Position")
If you and your GM think an attack is a single swing, then it's completely reasonable to declare that as your trigger. Recalling that Readying a spell includes the entire casting, and only a (presumably instantaneous) release of the effect.
But if you and your GM think an attack is more-complex, perhaps there'd need to be a perception or insight check to decide whether you're being targeted versus just observing the normal bob-and-weave of combat.
But there's another level to consider: targeting, discrete attacks, triggers... these are all just abstractions that allow us, the players, to quickly narrate the actions of characters without having to act it all out in real time. For some it's entirely reasonable to say "a character trained and experience in combat can recognize when an attack's coming their way."**
Just as Alice's declaration of "I target Bob with my..." is our shorthand for all the movement Alice does to get in position, lining up weapon, feint..., "my spell will trigger if Alice targets me" can be our shorthand for all the movement Alice does to get in position, line up her weapon, fake me out....
that you and your table decide together to take the metagame/abstract approach. 5e just isn't trying to be so super-fine in its details. Unless you want to zoom to the level of declaring every second of your characters' lives---which level of magnification 5e doesn't really support---then let meta-actions be valid triggers.
* - I have made it a practice in the PHB to interpret the target of "you" anew on a paragraph-by-paragraph basis. That's usually useful, though there remain plenty of examples where this obviously breaks down.
** - True story: the one bar fight I've been in I was literally yanked out of a split-second before a punch flew past my face. I'd never been in a bar fight before that. I'd never been in any fight before that. I had no idea the fight was starting--nay, had started--when the guy to the right of me wound up to punch the guy to the left of me.
My companion, more experienced than me, saw the targeting happening. I did not. My companion, luckily, was quick enough to yoink! me to safety. Thanks, Cutler.
There will be situations not covered in the rules, in every game. The purpose of the DM role is to represent the game world itself. To do that, DM uses a bunch of tools, including the rules. That's the major difference from computer role-playing games.
"Whatever the DM says, goes", even when it contradicts the rules - is known as "Rule Zero" or "The Golden Rule". DM simulates reality of a fantasy world and have to use their common sense, in addition to the rules and/or the plot.
What situation we are trying to simulate here? Two men confront each other. The crossbowman aims the mage. The mage tries to teleport away.
The possible options:
Option 1 totally depends on the crossbowman intentions. Options 2 and 4 are basically the same situation, where the result depends on who is faster. Which has already been decided through the initiative roll.
Now the question is - what’s the likelihood of the 3 option?
The Teleport spell has a verbal component and a casting time of one action. At least a couple of seconds (up to six) the mage have to chant mystic words in order to start the teleportation. It seems impossible to do that the very same moment the crossbowman pulls the trigger.
However, the Ready action description says:
When you ready a spell, you cast it as normal but hold its energy, which you release with your reaction when the trigger occurs.
So, you have to cast the spell in advance, and then release it as a reaction. That seems more plausible in our "crossbowman vs. mage" scenario. Still, it isn't easy to react right before your enemy pulls the trigger, until you are extremely lucky. You said "if he aims me", but he could aim (but not shoot) as a free action, not as a part of an attack.
Assuming it's the matter of luck, how would I resolve the situation: Ask the mage player to roll d20. On 19 or 20 it will be option 3. Otherwise, it's option 2.