Short answers, pulled from the story.
The Kurgan hypothesis is the most widely accepted theory identifying the Proto-Indo-European homeland. It proposes that speakers of Proto-Indo-European lived in the Pontic-Caspian steppe north of the Black Sea and spread their language into Europe and parts of Asia during the 5th through 3rd millennia BC.
The steppe theory was first formulated by Otto Schrader in 1883 and V. Gordon Childe in 1926. It was then systematized in the 1950s by Marija Gimbutas, who introduced the term "Kurgan culture" in 1956 in The Prehistory of Eastern Europe, Part 1.
The first strong archaeological evidence for horse domestication comes from the Sredny Stog culture north of the Azov Sea in Ukraine, dating to the 5th millennium BC. In the Kurgan model, the domesticated horse and early chariots provided the mobility that allowed steppe populations to expand across vast distances.
The Yamnaya, also called the Pit Grave culture, is a cultural horizon spanning the entire Pontic-Caspian steppe from roughly the mid-4th to the 3rd millennium BC. Gimbutas identified it as Kurgan IV, the peak stage of expansion; David Anthony later used it as his primary point of reference in his 2007 revised steppe theory.
Genetics studies confirmed that populations bearing specific Y-DNA haplogroups and a distinct genetic signature expanded from the Pontic-Caspian steppe into Europe and South Asia during the 3rd and 2nd millennia BC. The evidence supports a migratory origin for at least some Indo-European languages and suggests the population movement may have been larger than earlier models assumed.
The Anatolian hypothesis places the Proto-Indo-European homeland in Neolithic Anatolia rather than the Pontic steppe. Genetics findings are generally considered to favor the steppe theory; the Anatolian hypothesis is regarded as incompatible with the linguistic evidence, even though researchers like Alberto Piazza and Luigi Luca Cavalli-Sforza attempted to align the two theories in the 2000s.