Approximately half of the people on Earth speak an Indo-European language. Some of the expansion of this language has taken place in historic times through European conquests, but most of it occurred before the dawn of history. Most of Europe and much of Asia were speaking IE languages before historic times.
J.P. Mallory, in his book In Search of the Indo-Europeans, begins his chapter on the search for the Indo-European homeland by quoting three separate declarations by a single authority, spaced over 47 years, confidently assigning that homeland to Asia, Europe, and Asia Minor respectively. Nonetheless, Mallory remains confident that the IE homeland has already been identified, mainly because essentially every semi-plausible (and many an utterly absurd) potential location has already been claimed by somebody. In the absurd crowd, I would count the North Pole and Iceland.
One complicating factor is nationalism and racism. Mallory also devotes a chapter to the Aryan Myth, which reached it's moral nadir in Nazi Germany but flourished for many decades earlier. One component insisted that the original IE speakers, or Aryans, had to be blue-eyed blonds from Northern Europe. This hypothesis, I guess, was popular among blue-eyed blond Northern Europeans. A more modern, and slightly better founded notion is the out of India hypothesis, which in one form, claims that the IEs were from the Harrapan civilization of the Indus valley. A crucial defect of this theory is that the Indo-Europeans were very horse-centric, and the Indus Valley Civilization seems to have been nearly without horses.
Motivation for such out of the main stream ideas comes at least partially from the desire of cultures to claim autochthony, the notion that their culture is uncontaminated by foreign ideas, or, at least, that it can be purified of such foreign notions and be better for it. This impulse to claim autochthony was strong in Germany of the dawn of the Twentieth Century, and it is at the center of modern Hindu nationalism as well as Muslim extremism, not to mention our own Christian fundamentalism.
I mentioned this to an archaeologist friend and his reaction was "There is no such thing as an autochthonous civilization." Ironically, the claim seems to have its strongest appeal to those for whom the claim is most tenuous, like the long divided and frequently conquered Germans, Indians, and Arabs. There is no doubt that such myths have immense power, though, and many of the most successful conquerors, like the Romans, were eager to attach themselves to a more ancient parentage, which the Romans found in Troy.
So, about those Indo-Europeans? Will we ever sort out from whence they came? Until recent times, this was purely a domain of archaeology and linguistics, but in recent decades the plummeting cost of DNA analysis has made archaeogenomics the principal player. The most plausible model still seems to be the notion that the ancestral homeland was in central Asia, but those people may themselves have emigrated from somewhere else, including perhaps Anatolia or even the Indus valley.
The clearest signals appear to come from Europe, where ancient DNA has occasionally been preserved. There is evidence of at least three genetically distinct waves of settlement, first, hunter-gatherers, next, neolithic farmers, and finally, peoples related to cultures of the steppe, who quite plausibly might be the Indo-Europeans.
India remains murky, partly because of the genetic complexity of the subcontinent, which has been a way station for nearly every migration since we left Africa, but also because the climate does not favor the preservation of ancient DNA.
TBD - or, maybe, not.