Haüyn was first discovered by Abbé Gismondi at Lake Nemi (Italian: Lago di Nemi) in the Latium region of Italy and described as Latialite in an unpublished mineralogical treatise in 1803. He passed his treatise to the Danish scholar Tønnes Christian Bruun-Neergaard (1776–1824), who used it as the basis for a lecture in the class of sciences at the National Institute on May 25, 1807, to introduce the new mineral to the public. As it was mostly rejected at the time to name minerals after their first place of discovery, and the new mineral had already been found on Monte Somma, Bruun-Neergaard suggested naming the new mineral after the French mineralogist René-Just Haüy (1743–1822) as Hauyn (now correctly spelled: Haüyn).

Haüyn (pronounced [ha'ɥi:n]), also known as Hauyn or formerly as Hauynite, is a relatively rare mineral from the mineral class "Silicates and Germanates" with the composition Na5-6Ca2[(SO4,Cl)2|Al6Si6O24], chemically a sodium-calcium-aluminosilicate with [SO4]2− and chloride as additional anions.

Haüyn crystallizes in the cubic crystal system, but usually forms only millimeter-sized crystals, predominantly blue in color with a glassy luster. In rare cases, white, brown, yellow, gray, green, greenish-blue, and orange-red Haüyne have also been found.

Haüyn forms in SiO2-undersaturated (silica-poor, alkaline) volcanics during the late stages of magmatic differentiation. During explosive volcanic eruptions, it is ejected from the depths of the Earth's crust. The fallout contains volcanic ash, pumice, tuff, and slag. Associated minerals include titanian andradite, apatite, augite, biotite, leucite, melilite, nepheline, phlogopite, and sanidine.

"Gem-quality," meaning suitable for the jewelry industry and facetable, Haüynes are mainly found in the ash and pumice layers covering the surroundings of Lake Laach near Mendig and Nickenich in the Eifel region. The largest known Haüyn crystal, approximately 3.2 centimeters in diameter, was found in early October 2012 by an amateur collector in the Eifel and has been owned by the TU Bergakademie Freiberg since early 2013. Currently, the crystal is on display at the Krüger-Haus in the exhibition "German Minerals" of the "Mineralogical Collection Germany" foundation.

Other well-known locations in Germany include besides the Eifel, Hochkopf and Hirzberg in the Black Forest, as well as several sites in Kaiserstuhl in Baden-Württemberg, and the Roßberg quarry near Roßdorf in the Hessian Odenwald.

In Austria, Haüyn has only been found at Pauliberg and Stradner Kogel, and in Switzerland, the mineral is known only from Beringen SH and Reiat in the canton of Schaffhausen.

Worldwide locations include, among others, Badakhshan in Afghanistan, Armenia, the island of Tasmania off Australia, Itaju do Colônia in the Brazilian state of Bahia, Sumaco and Pan de Azúcar in the Ecuadorian province of Napo, Nunavut and Quebec in Canada, Auvergne-Rhône-Alpes in France, as well as the French colony Tahiti, Kangerlussuaq in Greenland, the Los Archipelago in Guinea, various regions in Italy, the northern regions of Russia, Dalarna and Uppland in Sweden, the Canary Islands, and Catalonia in Spain, as well as several states in the USA.

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