We now have clear evidence that the Indo-European languages spread to Europe in a wave of pastoral conquest 4000 - 5000 years ago. Those of European extraction (like myself) are descended from a mixture of these people with the earlier hunter-gatherer peoples and farmers. These conquerors originated in the Yamanya culture to the North and East of the Black Sea. The Indo-European languages also spread to India, Anatolia, and parts of China. Scholars have argued for a century or more about the ultimate homeland of the I-E peoples, with every semi-plausible theory and numerous silly ones being embraced somewhere, by someone.
Most today would pick the Yamnaya culture and its location, but there is a significant minority who argue for an Out-Of-India (OOI) origin. In my opinion this argument is driven more by nationalistic fervor than evidence or logic, but truly conclusive evidence is so far lacking. For Europe, the evidence came in the form of DNA from ancient skeletons which allowed genetics to trace the movements of the relevant peoples.
The favored origin of the OOI theory is the Indus Valley Civilization which thrived at roughly the same time as the Yamnaya. For the first time, a systematic investigation of skeletal remains at Rakhighari, one of the large cities of the Indus Valley Civilization, seems to being yielding human DNA. This could conclusively settle the question of whether or not this civilization was genetically similar to the Indo-Europeans.
Stay tuned. Rumor has it that results will be presented at the Eighth World Archaeological Conference, 28 Aug to 2 Sep 2016, in Kyoto.