Old Ones
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Old Ones is a term used in fantasy games for the culture that was first on the scene in creation, before the limits of recorded history in the current setting of the game. This culture is always described as awesomely powerful, responsible for shaping entire landscapes, sinking continents beneath the waves, or giving birth to new races. An important feature of a race of Old Ones is they aren't around anymore -- or if they are, it's in some degenerate form (reduced to stone-age beasts, or secluded colony hiding from the world, or one sleeping god-like individual).
Precursors is the sci-fi version, but doesn't carry the connotation of malevolence.
Call of Cthulhu
There were many "Old Ones" in the Cthulhu Mythos, and thus the Call of Cthulhu RPG. It's most commonly used to refer to one of the following two:
The Great Old Ones were alien beings of colossal size, and not exactly normal physics. All the Great Old Ones have been slain or imprisoned by Elder Gods, some entombed in planets like Earth. Some theorize that the Great Old Ones are imprisoned for the crime of practicing blasphemous black magic, and the Great Old Ones are just waiting to break out and avenge themselves. Others theorize the Great Old Ones submitted willingly to imprisonment, as a way to sleep away the aeons "until the stars are right" and they can resume their awful practices.
The race of Old Ones (or Elder Things) were extraterrestrials that traveled between the stars, and set up a colony on primordial Earth. They built a city on the southern polar continent and experimented with the local biology to create a servant caste. Their discards were tossed to the other continents (the Cambrian Explosion? The Cambrian Explosion pre-dated life on land.) and settled on their ideal lowest-class citizens: the Shoggoths, amorphous blobs of eyes and toothy mouths. The Shoggoths formed a rebellion and overthrew their masters, imprisoning them in suspended animation tombs of ice sealed with Elder Signs.
Forgotten Realms
Long long time ago, a lizardman type race had a huge empire that enslaved all the "warmbloods." It was probably the Sarrukh, mentioned in the Serpent Kingdoms splatbook. They're supposed to be extinct now, but it would make a decent BBEG. The Story of the First NeverWinter Nights explored this but made it all very confusing and disappointing.
Traveller
The old-ones in Traveller are called the "Ancients," and their sites can be found in any random place. The architecture can vary wildly, from habitrail tunnels hermetically sealed and protected from a benign environment, to hollowed-out crystals hundreds of feet high with no ladders or stairs. None of the sites have any documentation, many of them expect visitors to be able to fly or use telepathy, and all of them are empty and inert... except maybe whatever one your adventurers carelessly step into . Each Ancient site predates the literacy of every civilization, and doesn't seem to have anything in common with other Ancient sites other than being built for inhabitants that are 1-2 metres tall. Ancient sites make for good dungeon romps in what is otherwise a hard sci-fi campaign about making mortgage payments on your spaceship.
The Ancients also explain why there's so many humans in space. The first great human empire was the Vilani empire, built by the humans that first discovered faster-than-light travel. They were pretty surprised to find out there were other aliens around; even more surprised when they encountered another FTL space-faring empire of humans called the 'Zhodani.' Scientists concluded sure that the two-arms & two-legs body shape was just naturally better for most environments, and stopped being surprised that everyone was "humanoid" shaped. This all went to shit when a third race of humans showed up with FTL drive, calling themselves the 'men of Sol' ("Solomani" in Vilani), and it turned out that they came from a world where the entire ecosystem was integrated, not evolving in parallel like everywhere else. Vilani scientists were certain this was a sign that the Solomani were an engineered race of humans made by the Ancients, what with everything fitting together like clockwork. Instead, they discovered the opposite: the undeniable conclusion that Zhodani humans, Vilani humans, and all the other humans from minor races were "seeded" from Sol-3 and transplanted to every other world by the Ancients. Even the biology of the lupine Vargyr major race were seeded from this 'Earth' world. This gives the Solomani a superiority complex that makes them pretty insufferable to be around.
Warhammer Fantasy
Psychic frog men from beyond the stars; they put a warp-gate at each of the poles and shifted the planet's orbit to something nicer for their amphibian biology. All that's left of them in present day are the degenerate & decadent rulers of the Lizardmen armies, the Slann.
In older editions of 40K and WFB, these were the same Old Ones as the Old Ones below. In fact, the newest edition of the WFB rulebook specifically mentions their big silver space-ships, suggesting that that's still the case.
Warhammer 40,000
The Old Ones were a race or group of races* that were among the very first races in the galaxy, if not the first race(s) in the galaxy.
They were incredibly potent psykers (especially since the Warp was calm back in those days), had access to the Webway (and may have built or expanded it), and had supremely advanced genetic engineering techniques. They created the Eldar, Krork (likely the ancestors of modern Orks), the earliest ape ancestors of humans, Jokaero, the Hrud, the Rashan, the K'nib, and many others; some of these were created solely as warrior-races, whereas many (like the human predecessors) were created merely as part of planetary ecosystems. They were also immortal, a gift they seem to have given in some form to some of their creations.
However, it's basically impossible to talk about them without talking about the Necrontyr, a race of short-lived creatures obsessed with technology. The Necrontyr had grown up beneath a wretched sun that poured radiation upon them, so their lives were, without exception, incredibly short (although it's not clear how short). However, they managed to scrape together a small interstellar empire, with different start systems ruled by different dynasties. However, with the First War of Secession, the empire fell into infighting, as the dynasties violently jockeyed for power.
The Triarch (the ruling powers of the Necrontyr) decided a war was just what the doctor ordered to keep the Necrontyr together, and so declared war on the Old Ones, claiming the casus belli of "you didn't share immortality with us poor, pitiful, mortal creatures." This was, as would be expected, a popular cause, and started the so-called War in Heaven.
However, despite having much greater numbers and mastery of technology which matched or even exceeded that held by the Old Ones, the Necrons did not have access to the Webway, and so were constantly outmaneuvered. Soon, the Necrontyr were scattered and left with only a few coreworlds to the galactic north. With the War in Heaven collapsing, the Second War of Secession began, and the Necrontyr began to tear themselves apart again.
Before Necrontyr society collapsed entirely, however, the Necrontyr discovered the C'tan, the Star Gods, leaching on some of their stars. The Necrontyr made metallic bodies for the C'tan, which barely contained their incredible power, as they were basically material universe's equivalent to the Chaos Gods.
The C'tan offered the Necrontyr a solution: living-metal bodies that were nearly indestructible. For some reason, the Necrontyr were also renamed the Necrons at this point. One of the C'tan, known as the "Burning One," also helped the newly-robotic Necrons enter the Webway using the Dolmen Gates. With the field leveled, the Necrons were now able to take the fight to the Old Ones.
At this point, the Old Ones created many of the aforementioned warrior races, especially the Eldar and the Krork, but even these couldn't stop the Necrons. The wild misuse of the Warp had also spawned the Enslavers (or, at least, let them enter the material universe), obliterating many of the Old Ones. (The birth of Chaos caused by the corruption of the Warp by the dying emanations of countless warrior races probably didn't help either, to say the least. Incidentally, this also means that the current state of Grimdark in the galaxy is all their fault.) Weakened, the Necrons and their C'tan allies were then able to annihilate the rest of the Old Ones, at least according to Codex: Necrons 5th Edition. After that, the Necrons realized that losing their souls was a bad thing and turned on the weakened C'tan; while they were successful in breaking them into countless "shards" of their former power and permanently killed at least one, they feared the retaliation of the Eldar and the other surviving warrior races and went into hibernation until they were ready to rebuild their empire.
However, although most races aren't aware of their Old Ones heritage, they still made a great impact on the nascent races. In particular, at least according to Xenology, many races, including the Eldar, have very similar gods, including a bloody-handed warrior, a great smith, a life-giving goddess of the harvest, a laughing god, etc. It seems that these gods were inspired by the Old Ones, although whether they're based on particular Old Ones or just general concepts of what they did is uncertain.
Also, in Hrud mythology (again, according to Xenology), there's one last Old One named Qah that only left 500,000 years ago (i.e., long after the Necrons entered hibernation), promising to return when the Hrud reunite to fight the C'tan again. This would imply that, in fact, the Old Ones were not destroyed entirely by the Necrons, and a few hid or escaped.
And then, of course, there is there role in the Fantasy universe. If Fantasy takes place around the same time as 40k (as is occasionally implied), this would mean the Old Ones were active only 7,000 - 10,000 years ago, which also implies that they were not all destroyed.
- Usually, the Old Ones are referred to as a single, specific race. However, Codex: Eldar 4th Edition, as well as some of the Deathwatch supplements, claim that they were actually a group of the earliest races, who all worked together.