Emigration to the Americas, Australia, Africa, India (and perhaps Europe and the Mediterranean):
The National Archives
The Catalogue
Ships' passenger lists exist only for the period 1890-1960. There are no passenger lists for air travel.
No lists exist for those emigrating to Ireland.
You will not find records of emigrants to Europe and the Mediterranean unless their ship was travelling elsewhere in the world and touched at a European port en route.
Online
You can now access outward passenger lists (held in the series BT 27) online, courtesy of findmypast.com. There is a fee to download the records, although they are free to access on site. Find out more about these records