I've always liked the idea that the Indo-European language family started in Kazakhstan. I know very little about Kazakhstan, other than that it boasts the world's first, largest, and surely busiest spaceport, Baikonur, and its flag is quite attractive, I think.
Most people put the Indo-European origin here or nearby because of the Kurgan Hypothesis that was proposed in the 1950's by Marija Gimbutas. The Kurgan peoples, she thought, spread "with fire and sword" through Europe starting between 4,500 and 2,500 BC. It is an attractive hypothesis for archaeologists because it ties the spread of the languages to the spread of a material culture.
Colin Renfrew, an English archaeologist, rose to challenge the accepted version with an Anatolian Hypothesis. It puts the starting point for Indo-European languages in the Anatolian Plateau of Turkey and the starting time much earlier, about 8,000 to 10,000 years BC. He tied the spread of the languages to the spread of farming or, in other words, the Neolithic Revolution. He thought that farmers migrated more or less peacefully to fresh farmlands and mixed with the local population.
I mention these theories because a BBC News article says that an article in the journal Science provides some strong support for Renfrew's ideas. The researchers used a mathematical approach called phylogenetics that was developed to trace the origins of a disease, but they used it on the Indo-European languages instead of viruses. The result suggests that the languages spread from Anatolia at about 6,000 to 7,500 BC.
I think that DNA evidence is also beginning to support Renfrew's ideas. A study of the DNA of the earliest German farmers (the original paper is here) suggests that farming groups migrated originally from Anatolia or the Near East, but their DNA is only a small percentage among Europeans now. They may have been only a minority compared to the original hunting population in northern europe, but that would not prevent them from influencing the culture (including the languages) out of all proportion to their numbers.
I expect that new evidence for and against the Anatolian Hypothesis is going to be published in the next few years. When a new consensus settles out, I'm going to read V. Gordon Childe's book The Aryans, A Study of Indo-European Origins (published 1926) and see how close he came, considering the comparatively limited information and techniques that he had.
A last, interesting, point: If you wonder what the Proto Indo-European language was like, the modern language that is the closest to it is Lithuanian.