The Indo European (IE) Languages are the most widespread in the world, now spoken virtually everywhere, but widespread in Eurasia more than 2000 years ago. The discovery of deep affinities between many of the major languages of India and those of Europe was one of the seminal events in linguistic history, but soon became ensnarled in racist and political controversies. The sequencing of ancient DNA from human fossils seems to have clarified the prehistory of Europe: an early population of hunter gatherers was largely but not completely replaced by neolithic farmers from the Iran and the Middle East, and they in turn were substantially replaced by mostly male pastoralists from Central Asia. These pastoralists are very likely to have brought the Indo-European languages.
The situation in India is more fraught, partly because many Indian nationalists are offended by the idea that their civilization might not be autochthonous, but more importantly because we don't yet have good ancient Indian DNA. The reason for the first circumstance is likely tied to the racist character of the original Aryan invasion theory, in which invading Aryans, presumed European, brought Vedic civilization to the presumed primitive Indians. This theory neatly fit into the narrative of the occupying British power. Of course we now think that the Aryans were not European, but Central Asian. Unlike Europe, however, India already had a complex civilization (the Indus Valley Civilization, or IVC) at about the time of the Aryan expansion.
An alternative to the notion that the IE languages came to the world from Central Asia is the Out of India theory, which argues that they had their origin in the IVC. This idea is popular in Hindu nationalist circles, but much less so among professional archaeologists - most of whom, of course, are western.
Razib Khan, writing in the July 27, 2017 India Today, has an excellent article on the state of play of the controversy.
A few highlights: 1)Ancient DNA results from the IVC site at Rakhigarhi are expected to be published this month. (2)Indirect evidence suggests that India, like Europe, had two waves of invaders, first farmers from Iran and the Middle East, and second, pastoralists from Central Asia, with the latter likely responsible for bringing IE languages and some elements later incorporated into Vedic culture. (3)As in Europe, the Aryan DNA was probably largely male.
It's possible that the ancient DNA from Rakhigarhi will testify in the dispute, but far from obvious that it would be conclusive.