The ancient Egyptians used a weapon called the khopesh. It was a curved blade that was excellent for getting around shields and puncturing body parts like kidneys.
Why did no other army in the ancient or medieval world later ever use the khopesh again, at least as a weapon type intended for specific enemies?
The khopesh was a solution to the limitations of bronze as a sword material.
Bronze swords can't be too long because they break. Bronze is more brittle and less flexible than iron. For this reason, bronze swords were used only as secondary weapons. The Greek xyphos was only used when the doris (spear) broke and some hoplites never trained with it.
The khopesh is a sickle sword. The curved blade has a very forward point of balance, making it similar to an axe (in fact, it evolved from axes).
It's an excellent hacking weapon, which makes it very useful to attack limbs not protected by armor or even break shields. It wasn't used "to get around shields" but through them, and neither "to puncture" because it is difficult to change direction between cuts, the point is almost blunt and backwards and the sword is too short for good lunges.
When swords began to be made with iron/steel, the extreme shape of the khopesh became less valuable. There are still "hacking" swords, like the Greek kopis (with a name so similar) and later the shotel or the scimitar, swords that favor the cut instead of the thrust. The form we usually call "machete" (falchion, etc.) has been popular in all history.
So, basically, ancient and medieval armies used the khopesh, but because it was made from steel, it didn't have its exact shape.
It was not because Bronze was softer or more brittle than iron as the accepted answer stipulates
The khopesh was mostly abandoned between ~1200-1100BCE which coincides nicely with the bronze age collapse, but the fact that it was historically made out of bronze has little to do with why the design was abandoned. Properly work hardened tin bronze can be tougher than mild steel (wrought iron) , and was not significantly bypassed by the properties of steel until the Medieval period when people started to figure out how to make medium carbon steels without too much sulfur and phosphorous contamination. Furthermore, not all bronze age swords were short. Many, like those used by the Minoan Cretians, had blade lengths in excess of 1 meter long.
The Bronze Age Collapse was the result of a devastating series of wars that began with the destruction of all the major Greek cities, and then spread to consume the entire Eastern Mediterranean. The bronze age ended, not because the iron of the day was better, but because these wars cut off the tin trade making bronze no longer a readily available alloy. Iron smelting was already discovered around 3000BCE long before the end of the bronze age, but bronze was the preferred metal for weapon making because the techniques available to make it weapon/armor ready were better.
It's also not because iron could not be used to make the same sort of shapes
While the exact shape of the khopesh was abandoned, the iron age saw several civilizations using equally recurved swords but with different edge profiles. The Dacian falx and Celtic sickle swords had nearly identical blades except that the cutting edge is on the inside of the hook instead of the outside.
So, if the absences of Bronze was not to blame, then the next most logical thing to look at is to see if the nature of warfare itself had changed.
It is unclear if the series of wars that ended the bronze age were caused by the Greeks themselves or not, but the one civilization that survived to leave us a written record were the Egyptians. In their descriptions and illustrations of the "Sea People" it is very clear that these attackers fought in a fashion very similar to the Mycenaean Greeks.
So, to understand the significance of these wars with the decline of the khopesh, you need to consider how the spread of Greek style warfare was fundamentally different from those regions that used the khopesh.
In the regions where the Khopesh was popular like Egypt and Canaan, most armies fought without any armor, but used large shields made out of wicker or a frame of wood covered in hide. Everything about the design of the Khopesh made it ideal for cutting through these shields. It was front heavy which gives it a lot of momentum when swung, but it also had a long, curved cutting surface so it could draw cut through these soft shields. There is also a kinesiological advantage to using a weapon that cuts in advance of the hand vs in-line with it.
In the Mycenaean Greek tradition of warfare, armor was much more popular. A panoply at the time would include bronze scale or even dendra style plate armor, a smaller but probably tougher shield, a boar-tusk helmet, a short straight sword, and a spear. While the Khopesh could cleave through hide and wicker just fine, cutting through metal armor (be it bronze or steel) is virtually impossible.
It was a curved blade that was excellent for getting around shields
This is a false assumption. Because the edge of the khopesh is on the outside of the curve, you have to turn your blade away from the target to hook a shield. Even when you do turn the khopesh backwards, the point does not lead the hand so it is not particularly easier to hit someone with around a shield than a straight sword. The point is also not inline with the thrust which means less penetration than a straight blade.
Against armor or a well made shield, a short, straight, well balanced sword is much more ideal because it is more maneuverable, and has a stronger more accurate thrust for getting into the armor's gaps or around the shield; so, when the sea peoples invaded the areas where the khopesh was common, the defenders switched to straight swords in response to seeing the advantages of them.
Throughout history, the general concept of the Khopesh would re-emerge over and over again in the form of the kopis, the scimitar, the cutlass, the kukri, etc. but it never again adopted the same questionmark like curve with an outer edge because every civilization moving forward knew that their blades might occasionally need to maneuver around a defense that would be too tough to just cut though.
Also used in Elam, Syria & Canaan (emphasis mine):
During the Middle Bronze Age the new sickle-sword spread rapidly through-out the Near East, appearing in Elam, Syria, Canaan, and eventually Egypt. Egypt seems to have been the last region to acquire the weapon. It doesn't appear in Middle Kingdom Egyptian art, making it likely that the weapon was initially acquired by Egyptians through trade or plunder from Canaan. There is mention of thirty-three "scimitars" - literally "reaping implements" (ECI 79 n49) - taken as plunder in Syria during the reign of Amenemhet II {l929-1895}. Presumably these are versions of the sickle-swords found in the royal tombs of Byblos in Syria and Shechem in Canaan during this period. The weapon does not seem to have been manufactured in Egypt until the New Kingdom, when it frequently appears in modified form as the Egypt khopesh (khps), or scimitar, where the haft of the weapon is reduced to about one third and the blade extended to two thirds AW 1:206-7; FP51).
Hamblin, 2006. Warfare in the Ancient Near East, p.71
