How does Overkill work in Hearthstone?

From the Hearthstone wiki (emphasis added):

Overkill is an ability that activates whenever the card with the keyword deals damage to a character exceeding its current Health during your turn.

It was introduced in the December 4, 2018 expansion.

As for strategies, I believe that that is a very broad question, but a player could use it to gain tempo by slightly Overkilling a minion to summon another, such as using the 5 mana Baited Arrow to kill another minion with 2 or less health remaining, to summon a 5/5 Devilsaur at the same time.

Ironhide Direhorn looks like another good card if it could survive a turn/attack immediately, perhaps when combo'd with Rush to let it Overkill a minion and summon a 5/5 Ironhide Runt.

Personally, I think Totemic Smash looks like it will give the best value for a Turn 1 play if the opponent plays a card with 1 HP; or it could also be used to remove another minion if you lack 1 HP to kill it with your other minions on board, in order to Overkill by 1 HP.