The Anatolian languages are a branch of the Indo-European languages which were spoken between the mid-4th millennium BCE and the mid-1st millennium CE in Anatolia. They are thought to have been the very first branch to break away from Proto-Indo-European, with some features not found in any other branch that are thought to have been archaicisms rather than innovations. Some of these include an animate/inanimate distinction for noun gender rather than masculine/feminine/neuter, and the retention of laryngeal consonants.
The best-attested Anatolian language is Hittite, from which we have records dating between 1650–1200 BCE. There are also some earlier records in the Akkadian language, from the 20th and 19th centuries BCE, which list some names and loanwords in Hittite. Hittite was the main language of the Hittite Empire, with the majority of records coming from that empire’s “royal city” of Hattuša. They were written in cuneiform script. Records end abruptly around 1200 BCE, when the empire fell.
The second-best-attested language in the group is Luwian, or the Luwic group generally. Luwian itself is attested in roughly the same era as Hittite, but other members of the sub-group are attested later, with short inscriptions attested up to the first few centuries CE.
Anatolia became heavily Hellenised following the conquests of Alexander the Great, which meant language shift in the area from Anatolian languages to Greek. They’re thought to have gone completely extinct around the 5th century CE, but had been fairly marginalised long before that.