The Wandering Days is the term used by Hobbits to refer to the centuries-long period of their westward migration from their races' ancestral home somewhere in the wild East (probably in the upper vales of the river Anduin).
Over many generations the Hobbits drifted westward until around Third Age 1050 many were settled until the Weather Hills, but many Stoors settled near Tharbad in Dunland. In T.A. 1300 the northern Hobbits had to flee from Angmar and some Stoors returned to the Gladden. The others continued westward and their first settlements were in Bree and Staddle.
Eventually, in the year T.A. 1601 a large population of Hobbits was granted permission by King Argeleb II, one of the least few Kings of Arthedain, to colonize the region beyond the Brandywine, and were later joined by the Stoors of Dunland. This would become The Shire, the main homeland of most of Hobbit-kind.
The oral history and memory of the Hobbits does not extend further back than their "Wandering Days", with their exact origins having been lost.