Wikipedia has a nice article on the history of the Romani people, sometimes called the Gypsies (a name based on an incorrect belief that they were a people who came to Europe from Egypt) or Roma (some use the term "Roma" to refer only to one group within the Romani people, I use it, perhaps incorrectly, but in line with some mass media conventions on the subject, to refer to all Romani people in this post). It marshals linguistic, genetic and historical sources documenting the argument, just on the cusp of the line between history and pre-history, that this ethnic group migrated from India to what is now Turkey (Anatolia), stayed there for a prolonged period of perhaps hundreds of years, and then migrated to Southeastern Europe.
A group of people known as the Dom, now numbering about 2.2 million in Iran, migrated from South Asia in the 500s, and may represent an earlier wave of migrants than the Roma. Isolated visits in the Byzantine empire by the "Atsingani", (a name sometimes attached to the Roma in the 1300s) are found from the 800s and before, but it isn't clear if they are the same group of people or not (it may have been a separate Manichean religious sect from Persia, i.e. Iran, or the Dom people).
The primary departure from India to the Byzantine Empire may have been a movement by defeated Northern Indian soldiers with their families in the wake of their defeat in raids by the Persian Sultan Mahmud of Ghazni into India around 1000 A.D. Mahmud's empire was an Islamic empire followed by the empire of the Seljuk Turks which gradually expanded West into Orthodox Christian Byzantine territory. Historical documents place the arrival of the Roma in Greece and Southeastern Europe in the 1300s, around the time that the Byzantine empire collapsed and was conquered by the Islamic Ottoman Empire (a successor to the empire of Mahmud of Ghazni and of the Seljuk Turks that preceded it).
Linguistic evidence suggests that the Romani language is most closely related to a branch of Hindustani languages that did not arise until about 1000 A.D. Their language is a descendant of Sanskrit rather than Latin. Roma cultural practices and beliefs also suggest the South Asian roots that they themselves claim.
Genetic evidence also shows strong evidence of a link to South Asia. At least 47% of Roma men and about 30% of Roma women have genetic markers otherwise found only in India. There is also genetic evidence of "founder's effects" associated with a small population bottleneck between the current population and source population that is consistent with the overall historical picture. There is also many genetic subgroups within Roma populations in Europe, due either to "(i) a genetically substructured ancestral population, where the old social traditions of strict endogamy have been retained and subsequent splits of the comprising groups have enhanced the original genetic differences; (ii) a small homogeneous ancestral population spawning numerous subgroups where strong drift effects have resulted in substantial genetic divergence."
The precise origins of the Roma within India are unclear. The general language family into which the Romani language fits is found in Northern and Central India. The cluster of genetic markers that is most common among the "Romanis is more prevalent in central India than it is in northern India." At least one genetic disease which is unusually common among the Romani, however, suggests a genetic link to the Jat people near Punjab, in Northwestern South Asia. The historical hypothesis that Sultan Mahmud's invasions led to the Roma migration would also point to a Northern India origin. One effort to synthesize the linguistic information suggests that the Roma's Indian ancestors may "have lived in the central Indian area, from where they emigrated to the north-west of India (about 250 BC) to reside there for a longer period of time. Experts still disagree on the point of time of the "gypsies‘" emigration from the northwest of India."
The founding population of Roma in Europe was small. According to one of the scientific journal articles cited by Wikipedia, the overall number of Roma in its Balkan provinces in the 15th century was estimated at only 17,000. There are now millions of Roma in Europe, where they are one of the continent's most ill treated minority populations.
In the case of the Roma, this represents a full circle migration. Linguistic evidence puts the origins of the Indo-European languages in Anatolia (present day Turkey) among some of the earliest farmers in human history around 8,000-9,500 years ago, not long after the neolithic revolution. One branch of the language family produced the Indo-European languages of India (e.g. Sanskrit and its descendants), another produced the Indo-European languages of Europe (e.g. Latin and its descendants). The Roma represent a group from the Eastern most wing of that language family (India) that migrated back into area where the Western wing of that language family took hold (Europe) through the area where the Indo-European languages probably originated (Turkey).
Some Ancient Jewish and Proto-Jewish History
In the case of the Jewish people, their historic language, Hebrew (which was brought back from the status of a "dead language" used only by scholars and for religious purposes into the living language of Israel), is a Semitic language rather than an Indo-European one.
The Semitic languages have their roots in the Middle East, North Africa and the Horn of Africa. The most widely spoken languages in that family today are "Arabic (322 million native speakers, approx 422 million total speakers). . . followed by Amharic (27 million) [in North Central Ethiopia], Tigrinya (about 6.7 million) [mostly in Eritrea], and Hebrew (about 5 million)." The Maltese language spoken in Malta, off the coast of Italy, is also Semetic and is the only Semetic language that is written in the Western alphabet.
The ancestors of Proto-Semitic speakers are now widely believed to have first arrived in the Middle East from Africa around the late Neolithic," although an alternative theory puts the source of Proto-Semitic in the Middle East itself, largely because some Semitic languages in Africa have Sumerian loan words (and Sumerian clearly did originate in the Middle East). Semitic languages spread out from "the Arabian Peninsula by approximately the 4th millennium BC."
A (now dead) Semitic language adopted the cuneiform script of Sumerian (the first known written language of humanity used in Southern Iraq and spoke from at least the 4th millennium BCE) and replaced Sumerian "as a spoken language somewhere around the turn of the 3rd and the 2nd millennium BCE (the exact dating being a matter of debate), but continued to be used as a sacred, ceremonial, literary and scientific language in Mesopotamia until the first century CE (AD)." Semitic Cuneiform was later abandoned in favor of the Aramaic script. The Sumerian language is now dead and is not related to any other known language (i.e. it is a language isolate), except through some loan words in some Semitic languages. (Today, there are only six language isolates on the Eurasian continent, the best known being Basque and Korean).
Other notable members of this language family are the (now dead) Phoenician (once spoken in the Eastern Mediterranean) and its successor Aramaic whose script was adapted by Hebrew, Arabic and other other Middle Eastern languages including non-Semitic ones. "It was the day-to-day language of Israel in the Second Temple period (539 BCE – 70 CE), the original language of large sections of the biblical books of Daniel and Ezra, likely to have been the mother tongue of Jesus of Nazareth and is the main language of the Talmud." "Modern Aramaic is spoken today as a first language by many scattered, predominantly small, and largely isolated communities of differing Christian, Jewish and Muslim groups of the Middle East —most numerously by the Assyrians in the form of Assyrian Neo-Aramaic," numbering in all about 2.2 million people and is considered an "endangered language." Aramaic is the closest living language relative of Hebrew. The languages have a common origin.
The Roma and the Jews
The Roma and the Jews are among the only examples of mass migration of non-Roman ethnic groups into Europe from the East until a mostly post-colonial wave of immigration in to Europe in the last few decades (the Jewish diaspora began about 1900-2000 years ago). One reason that their histories are notable is because they provide a concrete interface between the modern world and the part of ancient history that pre-dates what many college students are taught in Western Civilization.
History from late Greek civilization and the Roman Empire onward is quite well documented and widely known. The roughly 3,500 years before that era starts to get fuzzy, and I certainly don't know it that well. Like most people educated when I was, I also don't know much about the pre-Colonial eras for much of the rest of the world. One of my long term interests these days is to get a sense of this era, and to better pin down what we know about our ancient history, from the pre-Greco-Roman era back to the Neolithic revolution and pre-Neolithic migrations of the human race.
Our lack of knowledge isn't entirely an accident. Before the Sumarian, nothing was written down. Many of the modern world's other links to pre-Western, pre-Arabic Civilization were deliberately wiped out in the late Roman Empire and the early Islamic empire. Also, it is surely no accident that the age of the world as measured by young Earth creationists closely coincides with the age of the world's written history.
Both the Roma and Jews were subject to persecution on and off for most of their time in Europe, culminating in the genocide of the Holocaust for each group. Their experience in the last two thousand years can also help us trace questions about ancient history by providing empirical examples of answer to key questions faced by those trying to piece together the ancient past.
How much of a cultural and genetic impact can a small group of people that migrates into a larger established culture have on the larger culture? How isolated can a population of people remain while co-existing with a larger population?
In the case of the Roma, one can observe this quite directly. In the case of the Jewish people, the influences through the common origins of Christianity and rabbinic Judaism have to be parsed from the subsequent impacts on European culture.