Jeter l'eponge

Literally to throw the sponge, jeter l'éponge is the French equivalent of the English expression to throw in the towel.  Both expressions mean to give up and admit defeat, and are used roughly in the same way. If anything, the expression is more commonly used in French.  It is slightly more emphatic than the synonymous expression baisser les bras - lowering one's arms.

Now, why would the French throw a sponge instead of a towel?

Origin

Both expressions trace their origins to the sport of boxing, and more precisely to "cornermen." These sideline handlers jump in the ring in-between rounds and wipe excess sweat and blood off their fighters' faces with a sponge or a towel.  They can also play a more decisive role in a match.  If a fighter can no longer fight and refuses to - or is unable to - yield, they can intervene by throwing a sponge into the ring.  This signals surrender and spares the fighter further injury.  From this act of giving up the expression jeter l'éponge was born.

Originally, and up until the mid-800's, both the English and the French were trowing in the sponge.  Considering that boxing had developed largely in Britain until then, it's even likely that the French expression was first introduced as a translation from the English.  What about the English towel, then?  Well, it managed to replace the sponge somehow, at least in Britain.  Could it be that one day, someone who desperately wanted to stop the fighting and couldn't get a hold of a sponge, grabbed a towel instead?  We'll never know for sure.

Interestingly, this change of object is reflected in Quebecquois French where you'll hear jeter la serviette or lancer la serviette.

Just like its English counterpart, jeter l’éponge has taken on a wider meaning and is used in everyday situations, not just when talking about athletes.

Examples

Les investisseurs ont jeté l'eponge faute de rentabilité.
Les investisseurs threw in the towel because of a lack of profitability.

Découragé par le résultat des sondages, le maire a décidé de jeter l'éponge et de ne pas se re-présenter aux élections.
Discouraged by poll results, the mayor decided to give up and not seek re-election.