The word's origins are obscure; it may be of Celtic, Norman (French) or Anglo-Saxon origin. The nearest Gaelic term is oge maidne, or "new morning." Large Hogmanay festivals are now held in Edinburgh, Glasgow and Stirling.
"Auld Lang Syne" is often attributed to the Scottish poet Robert Burns, but Burns was only transcribing and adding to an existing folk song. Translated from the Scots dialect into everyday modern English, "Auld Lang Syne" means "old long since." In Scottish folk tales, "In the days of auld lang syne" takes the place of "once upon a time," and "for auld lang syne" could be restated as "for old times' sake."
The big disco ball is a modern descendant of the fireballs traditionally swung on Hogmanay. This tradition has its roots in pagan bonfire nights, also observed at Samhain and Beltane. The light wards off evil spirits and encourages the days to get longer. You'll see men carrying a 20-pound fireball attached to a 5-foot pole in Aberdeen's southern neighbor, Stonehaven. The fireball's traditional name is the Clavie.
Hogmanay is also celebrated with the custom of first footing. The first person to set foot through your door on the new year should be a tall, dark male. (Blonds, associated with the Viking invaders, were bad luck.) Traditionally, he would bear gifts of coal, whisky and black bun (a really heavy Scottish fruit cake) or another kind of cake. In eastern Scotland's fishing communities, he may also bring herring. Coal meant the house would stay warm for the coming year, and food gifts meant the house would not go hungry.
(Hogmanay 2007 in Biggar - by David Hamilton, Creative Commons)
Clement A. Miles writes in Christmas Customs and Traditions, "Drinking is and was a great feature of the Scottish New Year Eve."
During the Protestant Reformation years when celebrating Christmas was discouraged, Hogmanay became the gift-exchanging day in Scottish custom. Scottish children chose Hogmanay rather than Christmas Eve to go door to door wassailing - a begging custom that will remind Americans of trick-or-treating. Welsh kids carry a gray mare, but the old Scottish wassailing custom is for the leader of each group of children to dress in a sheepskin.
Hogmanay has its own website. This one is specific to Edinburgh Hogmanay.
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