Anatolian hypothesis
The Anatolian hypothesis suggests that the speakers of Proto-Indo-European (PIE) lived in Anatolia during the Neolithic era, and it associates the distribution of historical Indo-European languages with the expansion during the Neolithic revolution of the 7th and the 6th millennia BC.
This hypothesis states that Indo-European languages began to spread peacefully, by demic diffusion, into Europe from Asia Minor from around 7000 BC with the Neolithic advance of farming ("wave of advance"). Accordingly, most inhabitants of Neolithic Europe would have spoken Indo-European languages, and later migrations would have replaced the Indo-European varieties with other Indo-European varieties. The expansion of agriculture from the Middle East would have diffused three language families: Indo-European languages toward Europe, Dravidian languages toward Pakistan and India, and Afroasiatic languages toward the Arabian Peninsula and North Africa. Reacting to criticism, Renfrew revised his proposal to the effect of taking a pronounced Indo-Hittite position. Renfrew's revised views place only Pre-Proto-Indo-European in the 7th millennium BC in Anatolia, proposing as the homeland of Proto-Indo-European proper the Balkans around 5000 BC, which he explicitly identified as the "Old European culture", proposed by Marija Gimbutas. He thus still locates the original source of the Indo-European languages in Anatolia around 7000 BC.
Reconstructions of a Bronze Age PIE society, based on vocabulary items like "wheel", do not necessarily hold for the Anatolian branch, which may have separated at an early stage, prior to the invention of wheeled vehicles. According to Renfrew (2004), the spread of Indo-European proceeded in the following steps:

The main strength of the farming hypothesis lies in its linking of the spread of Indo-European languages with an archaeologically-known event, the spread of farming, which scholars often assume involved significant population shifts. Research published in 2003 of "87 languages with 2,449 lexical items" by Russell Gray and Quentin Atkinson found an age range for the "initial Indo-European divergence" of 7800 to 9800 years, which was found to be consistent with the Anatolian hypothesis. Using stochastic models to evaluate the presence or absence of different words across Indo-European, concluded that the origin of Indo-European goes back about 8500 years, the first split being that of Hittite from the rest (Indo-Hittite hypothesis). In 2006, the authors of the paper responded to their critics. In 2011, the authors and S. Greenhill found that two different datasets were also consistent with their theory. An analysis by Ryder and Nicholls (2011) found support for the Anatolian hypothesis: 
Bouckaert et al. (2012), including Gray and Atkinson, conducted a computerized phylogeographic study, using methods drawn from the modeling of the spatial diffusion of infectious diseases; it also showed strong support for the Anatolian hypothesis despite having undergone corrections and revisions. Colin Renfrew commented on this study, stating that "[f]inally we have a clear spatial picture. " Bayesian analysis has been criticized on account of its inferring the lifespan of a language from that of some of its words; the idiosyncratic outcome of, for example, the Albanian language raises doubts about the method and the data. Linguist Andrew Garrett, commenting on Bouckaert et al. (2012), stated that "[t]here is bias in the underlying data that leads to an erroneous conclusion, and strong evidence that is ignored which still strongly supports the Kurgan hypothesis." According to David Anthony, "this type of model doesn't match the complex linguistic and archaeological evidence," stating that "[t]he study is an example of retrofitting evidence to a model, but the results of such a model are only as useful as the underlying data and assumptions. "
Linguist Paul Heggarty from the Max Planck Institute wrote in 2014: 
Chang et al. (2015) also conducted a lexicostatistical (and some glottochronological) study, producing results different from the results produced by Gray and Atkinson. This study instead supports the Kurgan hypothesis. Piggot (1983) states that PIE contains words for technologies that make their first appearance in the archaeological record in the Late Neolithic, in some cases bordering on the early Bronze Age, some belonging to the oldest layers of PIE. The lexicon includes words relating to agriculture (dated to 7500 BC), stockbreeding (6500 BC), metallurgy (5500 BC), the plough (4500 BC), gold (4500 BC), domesticated horses (4000–3500 BC) and wheeled vehicles (4000–3400 BC). Horse breeding is thought to have originated with the Sredny Stog culture, semi-nomadic pastoralists living in the forest steppe zone, now in Ukraine. Wheeled vehicles are thought to have originated with Funnelbeaker culture in what is now Poland, Belarus and parts of Ukraine. According to Mallory and Adams (2006), linguistic analysis shows that the Proto-Indo-European lexicon seems to include words for a range of inventions and practices related to the Secondary Products Revolution, which postdates the early spread of farming. On lexico-cultural dating, Proto-Indo-European cannot be earlier than 4000 BC. According to Anthony and Ringe (2015) the main objection to the Anatolian hypothesis is that it requires an unrealistically early date. Most estimates date Proto-Indo-European between 4500 and 2500 BC, with the most probable date around 3700 BC. It is unlikely that late PIE, even after the separation of the Anatolian branch, postdates 2500 BC, as Proto-Indo-Iranian is usually dated to just before 2000 BC. On the other hand, it is not very likely that early PIE predates 4500 BC, as the reconstructed vocabulary strongly suggests a culture of the terminal phase of the Neolithic bordering on the early Bronze Age. Many Indo-European languages have cognate words meaning "axle": Latin , Lithuanian , Russian: , and Sanskrit: . (In some, a similar root is used for the word "armpit": in Old English, in Latin, and in Sanskrit.) All of them are linked to the PIE root . The reconstructed PIE root gives rise to , Hittite , Latin and Sanskrit , all meaning yoke. Words for "wheel" and "cart/wagon/chariot" take one of two common forms, thought to be linked with two PIE roots: the root "move around" is the basis of the unique derivative "wheel" which becomes (wheel) in Old Icelandic, (wheel, circle) in Old Church Slavonic, (neck) in Lithuanian, , (wheel, circle) in Greek, / (wheel) in Pali and Sanskrit, and (wagon, chariot) in Tocharian A. The root becomes (wheel) in Old High German, (wheel) in Latin, (wheel) in Lithuanian, and (wagon, chariot) in Sanskrit. The idea that farming in Western Eurasia was spread from Anatolia in a single wave has been revised. Instead, it appears to have spread in several waves by several routes, primarily from the Levant. The trail of plant domesticates indicates an initial foray from the Levant by sea. The overland route via Anatolia seems to have been most significant in spreading farming to Southeastern Europe. A genetic study from the Universitat Autònoma de Barcelona (2015) favors Gimbutas's Kurgan hypothesis over Renfrew's Anatolian hypothesis but "does not reveal the precise origin of PIE, nor does it clarify the impact Kurgan migrations had on different parts of Europe". Lazaridis et al. (2016) noted on the origins of Ancestral North Indians:
However, Lazaridis et al. previously admitted being unsure "if the steppe is the ultimate source" of the Indo-European languages and believe that more data is needed.
Description
The Anatolian hypothesis, also known as the Anatolian theory or the sedentary farmer theory, first developed by British archaeologist Colin Renfrew in 1987, proposes that the dispersal of Proto-Indo-Europeans originated in Neolithic Anatolia.