Upvoting. Mainly for "Humanity's just getting around to writing decent documentation, though".

Note that your last paragraph is actually only half the picture. In fact, all species that are domesticated today where domesticated thousands of years ago, before any written records existed. Prehistoric humans already domesticated all species that are domesticated today.

@quarague With the exception of the Russian Red Fox....

@quarague: "All species that are domesticated today where domesticated thousands of years ago", of course excepting those species which were domesticated in historical times, some of them quite recently. Domesticated hedhehog, silver fox, mink, rat, mouse, canary, koi carp, rabbit...

I think that maybe you need to explain the difference between taming and domestication to people, unless you are using them as the same word like many others seem to do?

I would add only one criteria to this answer: for those animals that are desired as something other than pets their desireable traits must at minimum not decrease and ideally increase. Meat animals should not become scrawnier, work animals should not become weaker, etc. This isn't likely to be the case anyway, but selecting for tameness and the desired trait both just makes it more difficult, and some traits could hypothetically be mutually exclusive with tameness.

From wikipedia on red foxes social behavior: "Red foxes live in family groups sharing a joint territory. In favourable habitats... [s]ubordinate foxes may number one or two, sometimes up to eight in one territory." -- I still think a baseline level of "social complexity" is a requirement for domestication. The eager-to-please the leader instinct has to exist first so we can convince it we are the leader. THEN we start the selective breeding to bring that instinct to the fore.

The red foxes may be the first case where we've put modern scientific effort into deliberately domesticating animals. Dogs and cats et. al. took a lot longer because we were just kind of letting it happen in a more passive way, not necessarily controlling their breeding to produce more docile and friendly animals. (Certainly there has always been some level of selective breeding for this sort of thing, but it wasn't quite so rigidly controlled as this fox experiment.)

The assumption that because you can do it with Silver foxes you can do it with anything seems entirely unjustified. Especially given the huge number of species that humans could apparently have potentially domesticated, and in many cases did tame and/or keep, but never domesticated.

@JackAidley Agreed. I found it to be a large leap from "we've domesticated a redfox" to "hence, we can domesticate anything".

@JackAidley The assumption here seems to be that selective breeding can affect animal behavior - the Silver fox is used as a demonstration of principle. Do we really feel that the general concept of "selective breeding informs animal behavior" is a concept that needs a mountain of supporting evidence?

@IronGremlin: No, but the idea that you can functionally utilise that to produce usefully domesticated species is a massive leap.

@Demigan Domestication by definition means "controlability." If you want to consider that tameness, you're welcome to. My answer points out that the breeding program must accomodate the goals of domestication (i.e., controlability). For a horse that would have included it's willingness to bear a burden or pull a plow. For a fighting dog, that would be aggression in the face of anything not its master - but in every case there must be control, and that will always resolve to some form of tameness.

@JohnO I think you're straining at a gnat. The Red Fox weren't merely less agressive, they were bred for the purpose of human companionship. In other words, there was an increase along with the decrease. I believe (not being an expert) that there will always be both because few animals (if any) will ever have all the positive traits you could ever want and/or none of the negative. Personally, I think the point I make in my answer is as valid for a war horse as it is for a child's pet or a food animal.

@DarrelHoffman I believe you're correct, but I don't believe the reverse (that previously it was all just hit-or-miss) is true. Horse breeding documents, as unscientific by today's standards as they may be, have been around for centuries if not millennia. Just because the scientific method hasn't been available to codify the process (the point of my last remark in the answer) doesn't mean that humans weren't perfectly capable of observing the effects of their efforts.

@JackAidley Where did I make that assumption? My answer specifically states, not all can. Indeed, millennia of selective breeding demonstrates quite well that it works just fine (see for example, cats, dogs, horses, cows, goats, pigs, birds....) I believe you've done me a wrong, sir.

@CharleyR. Where did you read in my answer that I believe we can domesticate anything?

Dairy bulls come to mind. To breed for greater productivity, they also bring in traits that make the bulls much more aggressive... the two traits are linked and (at least as of now) can't be separated. Similar linkages can exist for nearly any trait that people might value. One might even call dairy bulls feral. This is a big issue for all domestication, only the modern world's obsession with pets is the exception (since the only trait desired is friendliness, it can't be mutually exclusive with friendliness).

@JackAidley : indeed. It is important what the species was before domestication. Living in strong family units or packs makes domestication likely much easier than in case of completely solitary animals. Social animals only have to redirect their already existing feelings of belonging and communication towards humans, which might be much easier compared to animals which had absolutely no social skills whatsoever.

@JoinJBHonCodidact "I expect the same rules apply to Hippos" and "Eventually, you get a domesticated species" give the impression that you believe that it is more likely than not. Yes, you did add a small caveat hidden in the second to last paragraph (I missed it first time, sorry), but this small caveat goes against the bulk of your answer.