In Polynesian mythology, Hawaiki (also rendered as "Avaiki" (Society Islands), "Savai'i", (Samoa), "Havai’i" (Reo Tahiti)) is the original home of the Polynesians, before dispersal across Polynesia.  It also features as the underworld in many Māori stories.
Anne Salmond states Havai'i is the old name for Raiatea, the homeland of the Māori. When British explorer James Cook first sighted New Zealand in 1769, he had Tupaia on board, a Raiatean navigator and linguist. Cook's arrival seemed to be a confirmation of a prophecy by Toiroa, a priest from Mahia. At Tolaga Bay, Tupaia conversed with the priest, tohunga, associated with the school of learning located there, called Te Rawheoro. The priest asked about the Maori homelands, 'Rangiatea' (Ra'iatea), 'Hawaiki' (Havai'i, the ancient name for Ra'iatea), and 'Tawhiti' (Tahiti). 
The Māori word Hawaiki figures in legends about the arrival of the Māori in Aotearoa (New Zealand). The same concept appears in other Polynesian cultures, the name appearing variously as Havaiki, Havaiʻi, or ʻAvaiki in other Polynesian languages. Hawaiki or the misspelling "Hawaiiki" appear to have become the most common variants used in English. Although the Sāmoans have preserved no traditions of having originated elsewhere, the name of the largest Sāmoan island Savaiʻi preserves a cognate with the word Hawaiki, as does the name of the Polynesian islands of Hawaiʻi (the ʻokina denoting a glottal stop that replaces the "k" in some Polynesian languages).
On several island groups, including New Zealand and the Marquesas, the term has been recorded as associated with the mythical underworld and death.  William Wyatt Gill wrote at length in the nineteenth century recounting the legends about ʻAvaiki as the underworld or Hades of Mangaia in the Cook Islands.  Gill (1876:155) records a proverb: Ua po Avaiki, ua ao nunga nei – 'Tis night now in spirit-land, for 'tis light in this upper world." Tregear (1891:392) also records the term Avaiki as meaning " underworld" at Mangaia, probably sourced from Gill.  There is no real contradiction in Hawaiki being both the ancestral homeland (that is, the dwelling place of the ancestors) and the underworld, which is also the dwelling place of ancestors and the spirits.
Other possible cognates of the word Hawaiki include saualiʻi ("spirits" in Sāmoan) and houʻeiki ("chiefs" in Tongan). This has led some scholars to hypothesize that the word Hawaiki, and, by extension, Savaiʻi and Hawaiʻi, may not, in fact, have originally referred to a geographical place, but rather to chiefly ancestors and the chief-based social structure that pre-colonial Polynesia typically exhibited. 
- He-kî Hau Maka: "He kaiga iroto i te raá, iruga! Ka-oho korua, ka-û'i i te kaiga mo noho o te Ariki O'Hotu Matu'a!
- Translation: "The island towards the sun, above! Go, see the island where King Hotu Matuʻa will go and live!"
Englert puts forward the claim that Hiva lies to the West of the island.  The name Hiva is found in the Marquesas Islands, in the names of several islands: Nuku Hiva, Hiva Oa and Fatu Hiva (although in Fatu Hiva the hiva element may be a different word, ʻiva). It is also notable that in the Hawaiian Islands, the ancestral homeland is called Kahiki (a cognate of Tahiti, where at least part of the Hawaiian population came from).
According to various oral traditions, the Polynesians migrated from Hawaiki to the islands of the Pacific Ocean in open canoes, little different from the traditional craft found in Polynesia today. The Māori people of New Zealand trace their ancestry to groups of people who reportedly travelled from Hawaiki in about 40 named canoes (waka) (compare the discredited Great Fleet theory of the Polynesian settlement of New Zealand).
Polynesian oral traditions say that the spirits of Polynesian people return to Hawaiki after death. In the New Zealand context, such return-journeys take place via Spirits Bay, Cape Reinga and the Three Kings Islands at the extreme north of the North Island of New Zealand. This may indicate the direction in which Hawaiki may lie.
Until the early 21st century recently [update], many anthropologists had doubts that the canoe-legends described a deliberate migration. They tended to believe that the migration occurred accidentally when seafarers became lost and drifted to uninhabited shores. In 1947 Thor Heyerdahl sailed the Kon-Tiki, a balsa-wood raft, from South America into the Pacific in an attempt to show that humans could have settled Polynesia from the eastern shores of the Pacific Ocean, with sailors using the prevailing winds and simple construction techniques.
But DNA, linguistic, botanical, and archaeological evidence all indicate that the Austronesian-speaking peoples (including the Polynesians) probably originated from islands in eastern Asia, possibly from present-day Taiwan.   From there they gradually migrated southwards and eastwards through the South Pacific Ocean. The common ancestry of all the Austronesian languages, of which the Polynesian languages form a major subgroup, as well as all Austronesian language families but Malayo-Polynesian, exist only in Taiwan, and thus support this theory.
The sweet potato, which is of South American origin, is widely cultivated in Polynesia. This suggests that some interaction between the Polynesians and the Amerindians of South America may have taken place. But the sweet potato may also have been introduced in later trade by Europeans, or other Southeast Asians, where it was widely adopted. No Polynesian crops were introduced into the Americas, and there is no evidence of Polynesian settlement there. Austronesian and Polynesian navigators may have deduced the existence of uninhabited islands by observing migratory patterns of birds.
In recent decades, boatbuilders (see Polynesian Voyaging Society) have constructed ocean-going craft using traditional materials and techniques. They have sailed them over presumed traditional routes using ancient navigation methods, showing the feasibility of such deliberate migration that make use of prevailing winds.
- Hiroa, Te Rangi (1964). Vikings of the Sunrise. New Zealand: Whitecombe and Tombs Ltd. p. 69. ISBN 0-313-24522-3. Retrieved 21 August 2010.
- Salmond, Anne (2010). Aphrodite's Island. Berkeley: University of California Press. pp. 227-228. ISBN 9780520261143.
- Polynesian Lexicon Project Online
- Gill, William Wyatt, 1876. Myths and Songs from the South Pacific. Henry S. King, London, pp 152–174.
- This meaning may be archaic or forgotten in the Cook Islands today. Buse (1996:90) in his dictionary Cook Islands Maori Dictionary with English Finderlist (edited by Bruce Biggs and Rangi Moekaʻ) has this entry: Avaiki, prop. n. Hawaiki, the legendary homeland of the Polynesians. I tere tū mai rātou mei 'Avaiki. They voyaged direct from Hawaiki.
- M. Taumoefolau, "From *Sau 'Ariki to Hawaiki". The Journal of the Polynesian Society, 105(4), (1996), 385–410
- Englert notes that the phrase "The island towards the sun, above" seems to mean that, seen from Hiva, it lay toward the rising sun. Sourced from http://www.rongorongo.org/leyendas/008.htm
- "Mitochondrial DNA Provides a Link between Polynesians and Indigenous Taiwanese," synopsis. Public Library of Science, July 5, 2005
- "The origin of the Polynesians". The Economist, July 7, 2005.
- Buse, J., Taringa, R., Cook Islands Maori Dictionary With English Finderlist, edited by Biggs, B. and Moeka'a R. (1996), 90. Canberra: The Australian National University.
- M. Taumoefolau, "From *Sau 'Ariki to Hawaiki". The Journal of the Polynesian Society, 105(4), (1996), 385–410.
- E. R. Tregear, Maori-Polynesian Comparative Dictionary ( Lyon and Blair: Lambton Quay), 1891.
- Hawaiki in Te Ara – the Encyclopedia of New Zealand