Part of an ongoing (if occasional) series of 5e Rules notes.

Questions about this pop up sooner or later in every campaign, as it’s an area that 5e either does fine or poorly, depending on which online forum you go to or  specific question you examine.

So what happens when instead of bashing each other with hunks of steel or eldritch energies, you get up-close and personal

You can attack to Grapple

This is discussed on PHB 195. Essentially you use one of your Attacks to make a skill contest:

• Attacker: roll Strength (Athletics)
• Defender: roll Strength (Athletics) or Dex (Acrobatics)

In other words, Grapplers need to be a lot better at that skill, or be pretty darned strong.

If you win, the target is Grappled (technically, has the Grappled condition, PHB 290).

Which, honestly, doesn’t do much.  It basically reduces Speed to 0, even if it has other features that increase its speed beyond base.

But a grappled target can still punch grappler, stab them, etc. They are more restrained by being within 5′ of their grappler (which gives them Disadvantage on spell attacks or ranged attacks), but melee attacks are just fine.

Grappling does keep the grappled target from running away until help arrives or until the target breaks the Grapple. They can try that by  taking their Action to roll Strength (Athletics) or Dex (Acrobatics) vs the grappler’s Strength (Athletics) again. 

The grappler can also slowly (half speed) drag a grappled target with them — or continue attacking them at close range (with no particular advantage, and with the same disadvantages).

Note:

• When you have Grappled someone, you are not yourself considered Grappled.
• When you have Grappled someone, you are using up one hand (the other hand is considered free for attacks, etc.)

Aside from breaking free of the Grapple through a contest, it can also be broken by:

• The grappler being Incapacitated.
• An effect knocking the grappled target out of the reach of the grappler or the Grappling effect, e.g., being knocked away by a Thunderwave.

What if more than one person at a time is Grappling you?

While executing a Grapple takes one Attack (of however many you have within your Attack action), escaping a Grapple takes an entire Action. If more than one person has you Grappled, you can only escape one Grapple per turn … which means tag-teaming Grapplers can seriously cause you a problem.

Monster and Spell Grapples and Restraints

This is where things get a little tricky. Or more straightforward. You decide. The Grappled and Restrained conditions are quite separate, but come up a lot in spell and monster attacks: e.g., a Giant Octopus grapples you with tentacles; a Web spell causes the Restrained condition (PHB 292).

• Just like with a Grapple, a Restrained creature’s speed becomes 0, and it can’t benefit from any bonus to its speed.
• Attack rolls against the Restrained  target have Advantage, and the their attack rolls have Disadvantage.
• The Restrained target also has Disadvantage on Dexterity saving throws.

(Note that the Grappler feat lets the grappler get Advantage on the Grappled target — but, more interestingly, it lets a grappler roll a second Grapple attack, which, if it succeeds, means both the grappler and the grappled target are both Restrained. Which actually seems kinda sucky.)

So, Dave, how do I do something useful with this?

Grappling by itself is kind of limited in what it can do. But with a little work, it can make an effective attack.  So  how do you grab an opponent and actually subdue him.

The vast consensus I’ve read on this is Grapple them (so they can’t get away) and then do a Shove attack (PHB 195, same Ability checks as the Grapple) to knock them Prone …

…  at which point they get the Prone condition’s effects (PHB 292), and as Prone:

• they can only crawl (but not if they’re Grappled and at speed 0!)
• or they can get up (but not if they’re Grappled and at speed 0!)
• they are at a Disadvantage to attack (Prone!)
• attacks on them have Advantage (Prone!)
• … and they probably have to use their whole Action to try to desperately break the Grapple.

People have created whole builds around this.

Are you, the attacker, also Prone at that point? A good question. Consensus seems to be “No, but describe what you are doing to your DM.” I’d think of it as the arm twist behind the guy on the ground; you’re enough on your feet that you are not considered Prone, and can defend attacks normally (albeit with one hand), etc.

Given that, as long as you maintain this, you can continue attacking the Grappled+Prone guy and other folk can attack them, too, with Advantage, and that should distract them from attacking you back (as they desperately try to break the Grapple).