What is a donk-bet?
A donk-bet in poker is when you take the initiative by betting out of position (OOP) against the aggressor of the prior betting-round.
Back then, when nobody knew better, every player that donk-betted was labelled as a Donk, which was an insult, used to degrade the capabilities of the opposing player.
Now people slowly realise that a game-theoretical-optimal (GTO) strategy includes some donk-betting.

When to donk in poker | GTO Strategies
We aren’t going to fantasise on when to donk, instead we are trying to approximate GTO-solutions working with a poker-tool called PIOsolver. Let’s start.
Let’s check the following scenario: Button open-raises, you call and check the flop, he continuation-bets, and you call. YOu are 50bb deep at 10/20 blind-levels.

The flop -7d-6h-5s
Flop | Donking-range
In the images below, you will find hands that are marked green which signalise suggested calls, and red-marked hands that are suggested raises.
As you can see, you should barely ever donk-bet on this flop, but you should sometimes, mostly with hands that are very strong but vulnerable, and with hands that are not strong enough to withstand a (big) bet by the aggressor but have some value (T8s, Q8, J8 etc).
Should i donk some Turns?
Let’s check the following scenario: Button open-raises, you call and check the flop, he continuation-bets, and you call.
The turn is where it gets fascinating, PIOsolver suggests that it is necessary to implement a quite wide donking-range.
Click through the slideshow to see how you are supposed to put together your donking-range for the turn.
What did we learn?
What we learn from working out when to donk with PIOsolver is: We should mostly donk with small sizes (25-35%), rarely on the flop and mostly if the turn-card favours our range.
All other possible blank turns, which I have left out in the image slideshow, are turn’s that we should NEVER donk-bet because they don’t favour us, but the opponent.
You should rarely donk on the flop
The donk-bet should mostly be used on favourable turns
Don’t donk blank turns that don’t favour your range