Armenian hypothesis

The Armenian hypothesis of the Proto-Indo-European homeland, proposed by Georgian Tamaz V. Gamkrelidze and Russian linguist Vyacheslav Ivanov in the early 1980s, suggests that Proto-Indo-European was spoken during the 5th–4th millennia BC in “eastern Anatolia, the southern Caucasus, and northern Mesopotamia”. 
Recent DNA-research has led to renewed suggestions of a Caucasian homeland for a 'pre-proto-Indo-European'. It also lends support to the Indo-Hittite hypothesis, according to which both proto-Anatolian and proto-Indo-European split-off from a common mother language "no later than the 4th millennium BCE."
Hypothesis.
Gamkrelidze and Ivanov presented their hypothesis in Russian in 1980–1981 in two articles in "Vestnik drevnej istorii". During the following years they expanded and developed their work into their voluminous book, published in Russian in 1984; the English translation of the book appeared in 1995. In English a short sketch of the hypothesis first appeared in "The Early History of Indo-European Languages", published in "Scientific American" in 1990. Tamas Gamkrelidze published an update to the hypothesis in 2010.
According to Gamkrelidze and Ivanov, the Indo-European languages derive from a language originally spoken in the wide area of Armenian Highlands, the southern Caucasus, and northern Mesopotamia. The Anatolian languages, including Hittite, split off before 4000 BCE, and migrated into Anatolia at around 2000 BCE. Around 4000 BCE, the proto-Indo-European community split into Greek-Armenian-Indo-Iranians, Celto-Italo-Tocharians, and Balto-Slavo-Germanics. At around 3000–2500 BCE, Greek moved to the west, while the Indo-Aryans, the Celto-Italo-Tocharians and the Balto-Slavo-Germanics moved east, and then northwards along the eastern slope of the Caspian Sea. The Tocharians split from the Italo-Celtics before 2000 BCE and moved further east, while the Italo-Celtics and the Balto-Slavo-Germanics turned west again towards the northern slopes of the Black Sea. From there, they expanded further into Europe between around 2000 and 1000 BCE.
The phonological peculiarities of the consonants proposed in the glottalic theory would be best preserved in Armenian and the Germanic languages. Proto-Greek would be practically equivalent to Mycenaean Greek from the 17th century BC and closely associate Greek migration to Greece with the Indo-Aryan migration to India at about the same time (the Indo-European expansion at the transition to the Late Bronze Age, including the possibility of Indo-European Kassites).
The hypothesis argues for the latest possible date of Proto-Indo-European (without Anatolian), roughly a millennium later than the mainstream Kurgan hypothesis. In this respect, it represents an opposite to the Anatolian hypothesis in spite of the geographical proximity of the respective suggested "Urheimat" by diverging from the timeframe suggested there by approximately 3000 years.
Reception.
Renewed interest.
Recent DNA-research (2015-2018) has led to renewed suggestions of a Caucasian homeland for a 'proto-proto-Indo-European'. It also has been proposed by some to lend support to the Indo-Hittite hypothesis, according to which both proto-Anatolian and proto-Indo-European split-off from a common mother language "no later than the 4th millennium BCE."
Haak et al. (2015) states that "the Armenian plateau hypothesis gains in plausibility" since the Yamnaya partly descended from a Near Eastern population, which resembles present-day Armenians. Yet, they also state that "the question of what languages were spoken by the 'Eastern European Hunter-Gatherers' and the southern, Armenian-like, ancestral population remains open."
David Reich, in his 2018 publication "Who We Are and How We Got Here", noting the presence of some Indo-European languages (such as Hittite) in parts of ancient Anatolia, states that "the most likely location of the population that first spoke an Indo-European language was south of the Caucasus Mountains, perhaps in present-day Iran or Armenia, because ancient DNA from people who lived there matches what we would expect for a source population both for the Yamnaya and for ancient Anatolians." Yet, Reich also notes that "...the evidence here is circumstantial as no ancient DNA from the Hittites themselves has yet been published." Nevertheless, Reich also states that some, if not most, of the Indo-European languages were spread by the Yamnaya people.
According to Kroonen et al. (2018), Damgaard et al. (2018) aDNA studies in Anatolia "show no indication of a large-scale intrusion of a steppe population", but do "fit the recently developed consensus among linguists and historians that the speakers of the Anatolian languages established themselves in Anatolia by gradual infiltration and cultural assimilation." They further note that this lends support to the Indo-Hittite hypothesis, according to which both proto-Anatolian and proto-Indo-European split-off from a common mother language "no later than the 4th millennium BCE."
Wang et al. (2018) note that the Caucasus served as a corridor for gene flow between the steppe and cultures south of the Caucasus during the Eneolithic and the Bronze Age, stating that this "opens up the possibility of a homeland of PIE south of the Caucasus." However, Wang et al. also acknowledge that the latest genetic evidence supports an origin of Proto-Indo-Europeans in the steppe, noting: 
Kristian Kristiansen, in an interview with "Der Spiegel" in may 2018, stated that the Yamnaya culture may have had a predecessor at the Caucasus, where "proto-proto-Indo-European" was spoken.
Criticism.
J. Grepin wrote in a review (1986) in the "Times Literary Supplement" the model of linguistic relationships is "the most complex, far reaching and fully supported of this century".
Robert Drews says (as published in 1988) that "most of the chronological and historical arguments seem fragile at best, and of those that I am able to judge, some are evidently wrong". However, he argues that it is far more powerful as a linguistic model, providing insights into the relationship between the Indo-European and the Semitic and Kartvelian languages.
David Anthony in a 2019 analysis also criticizes the "southern" or Armenian hypothesis (replying to Reich, Kristiansen, and Wang). He finds that the Yamnaya derived mainly from Eastern European hunter-gatherers (EHG) and Caucasus hunter-gatherers (CHG), and suggests a genetic and linguistic origin of proto-Indo-Europeans (the Yamnaya) in the Eastern European steppe north of the Caucasus, from a mixture of these two groups. Anthony argues that the roots of proto-Indo-European formed mainly from a base of languages spoken by Eastern European hunter-gatherers, with some influences from the languages of Caucasus hunter-gatherers. According to Anthony, hunting-fishing camps from the lower Volga, dated 6200-4500 BCE, could be the remains of people who contributed the CHG-component, migrating from the south-east Caucasus, who mixed with EHG-people from the north Volga steppes. The resulting culture contributed to the Sredny Stog culture, a predecessor of the Yamnaya culture. Anthony cites evidence from ancient DNA, that the Bronze Age Maykop people of the Caucasus (previously proposed as a possible southern source of language and genetics at the root of Indo-European), had little genetic impact on the Yamnaya (whose paternal lineages differ from those found in Maykop remains, but are instead related to those of pre-Yamnaya Eastern European steppe hunter-gatherers). In addition, the Maykop (and other contemporary Caucasus samples), along with CHG, had significant Anatolian Farmer ancestry "which had spread into the Caucasus from the west after about 5000 BC", but is little detected in the Yamnaya. Partly for these reasons, Anthony concludes that Bronze Age Caucasus groups such as the Maykop "played only a minor role, if any, in the formation of Yamnaya ancestry." According to Anthony, this, the absence of evidence of significant admixture (including of paternal genetic influence, often associated with language shift) from the south on the Yamnaya suggests that the roots of Proto-Indo-European (archaic or proto-proto-Indo-European) were mainly in the steppe rather than the south. Anthony considers it likely that the Maykop spoke a Northern Caucasian language not ancestral to Indo-European.