It is generally agreed that they come from somewhere in North India. For example, there was a large genetic study published in 2012. Here's a table comparing the numbers 1-10 in various languages (this is taken right from Wikipedia) in Hindi, Persian (the largest two Indo-Iranian languages), Romani (unspecified dialect), Domari, and Lomavren. Romani 7–9 are borrowed from Greek, which obviously indicates a step on their journey to Europe (remember, until ultimately defeated by the Ottomans in the 15th century, the Byzantine Empire ruled much of Anatolia).
| number | Hindi | Romani | Domari | Lomavren | Persian |
|---|---|---|---|---|---|
| 1 | ek | ekh, jekh | yika | yak, yek | yak, yek |
| 2 | do | duj | dī | lui | du, do |
| 3 | tīn | trin | tærən | tərin | se |
| 4 | cār | štar | štar | išdör | čahār |
| 5 | pāñc | pandž | pandž | pendž | pandž |
| 6 | che | šov | šaš | šeš | šaš, šeš |
| 7 | sāt | ifta | xaut | haft | haft |
| 8 | āţh | oxto | xaišt | hašt | hašt |
| 9 | nau | inja | na | nu | nuh, noh |
| 10 | das | deš | des | las | dah |
| 20 | bīs | biš | wīs | vist | bist |
| 100 | sau | šel | saj | saj | sad |
Not only linguistic and genetic evidence, but people have argued that Rroma ideas about ritual purity, for example, are related to are Hinduism (a distant relation to be sure, but related). Rroma people are difficult to study ethnographically (one consistent is a healthy distrust of outsiders, who by all accounts are generally viewed as ritually polluting), but there are consistent clear rules about ritual purity (for example, keeping the top and bottom half of the bottom separate--including washing the clothes separately). This one is hard to say exactly, and is more speculative, but in my book plausible. There are indications that Rroma art, especially decorative wood carving for buildings and wagons, but I don't think I've ever heard of a systematic study of this.
It is not agreed when they left, exactly where in Northern India they left from, or what groups they were part of (defeated warriors? musicians? a group expelled in the Muslim invasions? a caste group that lost status?). The 2012 genetic study estimates they left Northwest India 1,500 years ago. However, linguistic evidence suggests a later date, because Rroma dialects have features that did not appear in Indian languages until somewhat later (for example, Rromani languages have only two genders, but until about 1,000 years ago, languages in Northern India had three genders, suggesting that the Rroma only left after this change happened). Nor is it clear how (if at all) they relate to similar groups, the Dom/Domari (perhaps related to the Dom/Domba still in Northern India), and the Lom people of the Caucasus. Are these early splits from the Rroma, or a separate wave of migration from the same or similar people?
It's also unclear how the Rroma are connected with others groups who clearly have a similar life style but no origin in India, like the Irish Travelers (derogatorily referred to as "Pavees" or Pikeys") and the Yenish people. They do not share a common origin, but do share a common social niche. It's hard to separate out the groups in the historical record because to outsiders they behave the same and were often referred to identically in the historical documents we have (a lot of what we know about Rroma comes from anti-Roma laws). They were both traveling groups with carpenters and tinsmiths ("tinkers" is a common old term for both), both spoke a language that indecipherable to outsiders, but Irish Travelers, as far as I can tell, do not have similar ideas of ritual purity as Rroma do, for example, nor to the best of knowledge did they work as fortune tellers and mystics. This confusion continues up to the present day: the British TV show My Big Fat Gypsy Wedding features almost exclusively Irish Travelers, not Roma. The movie Snatch, as another example, had Shelta-speaking Irish Travelers, not Rroma speaking a dialect of Rromani (which is, as made clear above, an Indo-Iranian language). As has been pointed out, Gypsy is generally not the most appropriate term. For one, it's ambiguous, and can refer to a mix of groups (including, at times, even New Age Travelers). For two, it's an inaccurate exonym (name from the outside)--it's derived from the term "Egyptian" (or, rather, probably ultimately the Greek word for Egyptian). And for three, it's generally considered offensive. "Roma", "Rroma", "Rromani", "Rom", "Roma and Sinti" are generally preferred (see Wikipedia's Names of the Romani people). There's disagreement even within the larger Romani about what the proper general terms are. I generally use the double Rr form of "Rroma" or "Rromani" for no other reason than that people are less likely to think I'm discussing Romans or Romanians.
For people who want to read more, Ian Hancock is generally recommended, but his work focuses more on human rights (and linguistics) than origins. His We Are the Romani People, for example, is not really a pure work of history (nor does it intend to be), though I think many recommend it as an introduction. There are also several ethnographic studies of Rroma; while I've only read a few, the ones I've read only seem to get pretty peripheral access. Jan Yoors might be the exception, but I've never read any of his work. The other difficulty is that there do seem to be rather large variations within Rroma culture across time and, more importantly, space, and I haven't seen any attempt to systematically combined the ethnographic and secondary literature that gets at both continuity and difference (this work may exist! I'm not particularly well read in the subject--this was mainly just one summer of curiosity).
Are there any substantial populations of gypsies in the US?