Anšar and Kišar are a pair of primordial gods that, with very few exceptions, only occur in Enūma eliš, the so-called "Babylonian Creation Story." They can be considered as part of an invented mythology that had the goal of creating a new cosmology for the god Marduk.
In Enūma eliš, the "Babylonian Creation Story" (for recent translations see Foster 2005: 436-368; Lambert 2008: 37-59), Anšar and Kišar appear as primordial gods that belong to the eldest generation of the Mesopotamian pantheon. According to Enūma eliš Anšar and Kišar gave birth to the god Anu (Black and Green 1998: 34), whereas a late lexical list suggest that Anu was sometimes replaced with Anšar (Reiner 1996).
In the Neo-Assyrian period, Anšar was sometimes equated with the god Aššur (Ebeling 1928; Edzard 1965: 41), most likely to give Aššur more seniority among the pantheon by identifying him with primordial gods (Black and Green 1998: 38; ; Beaulieu 1997). In Neo-Babylonian Uruk, Anšar/Aššur were also equated with the god Anu, leading to Kišar's identification with Antu (Beaulieu 1997). Kišar is sometimes also understood as spouse of the constellation Gudanna (Lambert 1976-1980b).
A bilingual (Sumerian - Akkadian) literary composition from Uruk, dating to the Seleucid period, poetically exalts the goddess Inana/Ištar. In lines 19-20 of that composition Kišar is equated with the goddess Antu, wife of Anu (Hruška 1969).
There is no evidence for cult places or worship of Anšar and Kišar.
Their attestations remain restricted to first millennium Mesopotamia, although Enūma eliš was probably composed earlier. Anšar occurs in two Sumerian literary texts, Enlil and Sud (ETCSL 1.2.2) and A hymn to Haya for Rim-Sin (ETCSL 22.214.171.124), yet it is unclear who this deity may have been.
The iconography of Anšar and Kišar is unknown.
In Sumerian the names Anšar and Kišar literally mean "the entirety of heavens" and "the entirety of earth."
Nicole Brisch, 'Anšar and Kišar (god and goddess)', Ancient Mesopotamian Gods and Goddesses, Oracc and the UK Higher Education Academy, 2012 [http://oracc.museum.upenn.edu/amgg/listofdeities/anarandkiar/]