Later research has pointed to their origin among the Xiong-nu of central Asia, ranign from Siberia to Mongolis and Manchuria, as well as China, and has suggested that it may have been pressure from the rising Han empire in China which led them to begin a westward migration (driving other nomadic peoples before them) ending circa 200 years later with entry into modern Europe.
By the 2nd century writers speak of them in the areas of the Caspian Sea and Dnieper River and around 361 CE they appear to have been invited, by the Romans to settle in Pannonia. Moving further west, they displaced tribes from Germany, but unlike others of the nomad peoples, they did not seek to spread further westwards and the empire which they established (represented in modern thought most often by the figure of Atilla) straddled Europe and Asia, stretching from the Germanies to the edge of the steppes, and from the Baltic to the Caspian.
With the death of Atilla, his sons proved unequalt to the challenge of holding the empire together and Hundom was broken for good by their defeat at the battle of Nedao.
The Huns were especially noted for their use of the compound bow and for their possession of horses as a component in their migration. These two factors offered them military and diplomatic superiority over their contemporaries. They were not xenophobic, and were willing to take other tribal groups under their overall banner -- thus (inter alia) Alans, Gepids, Sarmatians, Slavs and Goths rode wtih the Hun horde.
Source: Adapted from the Wikipaedia article