AQ when you don't hit the flop

[ASaysFull]

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Most of the time you check behind here and fold to the river bet from BB if you don't improve.

Sometimes you'll want to bet this again and check behind on the river. Going too far with overcards is one of the first leaks you should be fixing in your game.

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I agree, just because you raise a hand with "big cards" like AQo, AJs, or AKo/s, doesn't mean you have to keep going if you miss the flop. In this case though you are heads up, so I think checking and folding to river bet, or firing again on the turn and checking the river if possible could both be correct. It depends of your read on the bad guy and what you think he could do.

[Dankenstein]

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Most of the time you check behind here and fold to the river bet from BB if you don't improve.

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Here it is probably ok because the pot is small making the reward not worth the risk ;however, I'm pushing this turn in a larger pot. Checking behind on this turn is giving your hand away and will allow a good opponent to steal from you on the river.

[Aaron W.]

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No reads on opponent.

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Bet the turn 100% of the time and check behind on the river (or fold to a donk) UI. Villain could have peeled that flop with a huge range of overcard type hands and you're likely best.

[vixticator]

grunch

Bet again for value, metagame, etc. If you make a habit of betting flop and checking turn then your play is transparent. Players peel this type of flop liberally and you need to make them pay for it. Going to be led into on river all day if you check.