Simple question, asked in the title. Can you grapple as an opportunity attack. 

Unfortunately, no. Grappling is done by taking the Attack action and an Opportunity Attack is not the Attack action, it is a separate thing.

As Ophidimancer said - it's unequivocal.

Grappling
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.

RAW, definitely not, as Ophidimancer & Ember have said.

However, in game if a PC with a free hand wanted to try this ("As he tries to get away I reach out and grab the back of his shirt"), I think I would allow it. However, I'd probably put them at disadvantage to do so, or give their enemy a bonus or advantage, or even both depending on the circumstances. It would be much trickier to do in the limited time of a reaction than during a full (planned) attack. If they succeeded and their opponent hadn't yet used their action, I'd also allow them to try to escape the grapple immediately and, on success, continue their movement away.

The problem with allowing someone to grapple with an Attack of Opportunity is that you're devaluing the Sentinel feat - because basically everyone has it at that point.

Possibly, but not by much. Grappling would be in place of an OA attack, would require a free hand, and would be difficult to accomplish (with probable disadvantage on your athletics check). So, sentinel is still a big advantage over OA grapple.

Normally, no. You would have to have a feature that specifically allows for it.

Simple answer, as noted by everyone already: nope.

Quote from UrthTheThoughtless >>

Possibly, but not by much. Grappling would be in place of an OA attack, would require a free hand, and would be difficult to accomplish (with probable disadvantage on your athletics check). So, sentinel is still a big advantage over OA grapple.

The issue with allowing a grapple on OAs isn't the difficulty... it's straight up something that just is not allowed on an OA. WotC thought this through... an OA is very specifically "one melee attack", not the Attack action. Grappling specifically requires the Attack action by default.

Whether Sentinel has an advantage or not is irrelevant; it is still devalued by allowing someone to grapple on an OA (speed reduced to 0) without investing in a class feature or feat to do so.

Fair enough. As everyone has stated, RAW and, most likely, RAI it's not allowed. Anything else would be a house rule.

Quote from Kotath >>

It really does feel like another artificial play balance rule. It would make much of the RL game of American Football impossible if it did not work in real life. Tackling under normal D&D rules would be nigh impossible.

 Or all the players are Readying an Attack action, and they will perform a Grapple with their Attack action.

So in the case of a Water Weird where as the attack includes being grappled. Since this AOO happens on the players turn, Does the player get a escape save and then if the player has not taken his/her action, can they attempt a chance to break free on that same turn? 

Quote from Bodarian >>

So in the case of a Water Weird where as the attack includes being grappled. Since this AOO happens on the players turn, Does the player get a escape save and then if the player has not taken his/her action, can they attempt a chance to break free on that same turn? 

 If a Water Weird use Constrict and hit you during an Opportunity Attack, you're grappled and restrained. On your turn, you then can use an action to escape, provided you succeed on a DC 13 Strength (Athletics) or Dexterity (Acrobatics) check.

Reading the OP, if I'm not mistaken, it seems like something you can do now under the 2024 rules:

Opportunity Attacks

You can make an Opportunity Attack when a creature that you can see leaves your reach using its action, its Bonus Action, its Reaction, or one of its speeds. To make the Opportunity Attack, take a Reaction to make one melee attack with a weapon or an Unarmed Strike against the provoking creature. The attack occurs right before the creature leaves your reach. See also chapter 1 (“Combat”).

Unarmed Strike

Instead of using a weapon to make a melee attack, you can use a punch, kick, head-butt, or similar forceful blow. In game terms, this is an Unarmed Strike—a melee attack that involves you using your body to damage, grapple, or shove a target within 5 feet of you.

Action	Summary
Attack	Attack with a weapon or an Unarmed Strike.

 

Attack [Action]: When you take the Attack action, you can make one attack roll with a weapon or an Unarmed Strike.

I thought this sentence made it possible: (blue color)

Opportunity Attacks

You can make an Opportunity Attack when a creature that you can see leaves your reach using its action, its Bonus Action, its Reaction, or one of its speeds. To make the Opportunity Attack, take a Reaction to make one melee attack with a weapon or an Unarmed Strike against the provoking creature. The attack occurs right before the creature leaves your reach. See also chapter 1 (“Combat”).

Quote from TarodNet >>

I thought this sentence made it possible: (blue color)

Opportunity Attacks

You can make an Opportunity Attack when a creature that you can see leaves your reach using its action, its Bonus Action, its Reaction, or one of its speeds. To make the Opportunity Attack, take a Reaction to make one melee attack with a weapon or an Unarmed Strike against the provoking creature. The attack occurs right before the creature leaves your reach. See also chapter 1 (“Combat”).

It does. Since Opportunity Attacks can also be unarmed strikes, you can Grapple or Shove as an Opportunity Attack.

Any time you make an Unarmed Strike, you can choose to Grapple or Shove: https://www.dndbeyond.com/sources/dnd/free-rules/rules-glossary#UnarmedStrike

Unarmed Strike

Instead of using a weapon to make a melee attack, you can use a punch, kick, head-butt, or similar forceful blow. In game terms, this is an Unarmed Strike—a melee attack that involves you using your body to damage, grapple, or shove a target within 5 feet of you.

Whenever you use your Unarmed Strike, choose one of the following options for its effect.

Damage. You make an attack roll against the target. Your bonus to the roll equals your Strength modifier plus your Proficiency Bonus. On a hit, the target takes Bludgeoning damage equal to 1 plus your Strength modifier.

Grapple. The target must succeed on a Strength or Dexterity saving throw (it chooses which), or it has the Grappled condition. The DC for the saving throw and any escape attempts equals 8 plus your Strength modifier and Proficiency Bonus. This grapple is possible only if the target is no more than one size larger than you and if you have a hand free to grab it.

Shove. The target must succeed on a Strength or Dexterity saving throw (it chooses which), or you either push it 5 feet away or cause it to have the Prone condition. The DC for the saving throw equals 8 plus your Strength modifier and Proficiency Bonus. This shove is possible only if the target is no more than one size larger than you.

Being able to use Grapple or Shove on an Opportunity Attack is a subtle, but fundamental, change to melee combat in 5ER. It allows melee characters options to control combat in a way they had not in 5E. "Tanking" might actually be a thing now when it really wasn't a supported play style before.

Quote from Ophidimancer >>

Being able to use Grapple or Shove on an Opportunity Attack is a subtle, but fundamental, change to melee combat in 5ER. It allows melee characters options to control combat in a way they had not in 5E. "Tanking" might actually be a thing now when it really wasn't a supported play style before.

 You still need a free hand, though, right? If so it’s not going to help a lot for sword-and-board types. And heavy weapon users will be able to hold on, but otherwise just stand there with a bad guy in one hand and a glaive in the other. So a dual wielder could drop one weapon and at least still be armed and able to attack with the other. But then they’re often going to be dex-based, and the attack to grapple them is using str. This is all assuming we’re not talking about a monk, which has its own things going on.

You’re right that it does at least open it up as an option, I just wonder how often it will be a good option.

Oh yeah you now can i stand corrected! Still stuck in 2014 ;)

You deserve an inspiration TarodNet