In a world (Creative Mode) I spawned some horses with the spawn egg. When I finished playing, i closed Minecraft and shut down the computer.
When I re-opened the world, there weren't any horses.
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I was near them and they were all tamed (also, they had a saddle and an armor).
I played from two different computers and I had the same error.
Hm, I dunno then, guess it is a bug, just wanted to make sure.
I can confirm they disappear if their chunk is removed, unlike all other passive mobs. I wonder if this implies they also randomly despawn when more than 32 blocks away? This might partly explain the disputed reports of failure to spawn naturally.

Is this still a concern in the current Minecraft version? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.
Confirmed. In a game on Creative, I spawned like 20 horses with a spawn egg on an island with a tall marker, flew off until the marker disappeared, then flew back again, and there was only one untamed horse left. A tamed donkey and a tamed horse left on an adjacent island were still there, however, suggesting this only (or at least mainly) affects untamed horses.
However! I spawned a new batch. Saving, closing the world, opening another one and mucking about a bit, quitting out of Minecraft, and then restarting it and returning to the world with the horses, they were still there. Flying away until the chunks unloaded again, the horses once more disappeared, but the nearby 4 cows and 2 pigs remained. So it seems to be a chunk-loading thing, not a Minecraft-restarting thing.

Duplicate of MC-13716
Found a huge number of horses and mules. Built a corral around one of them. Logged off, and when I came back, no horses. The one in the corral was replaced by a chicken. Have not been able to find a single horse in Minecraft since then. Re-downloaded launcher. No luck. Tried generating a new world. Still no horses.
How far were you from them? Were they tamed? If you were far off and they were untamed, you logged out and logged in and went back to them and they were gone, that is normal; they despawn like hostile mobs if untamed.
I spawned some Horses and stayed near them, logged out and got off Minecraft, rebooted it up and logged into my world and they were there still. So you may wanna give a bit more info. 😞