I have been following the Hastings Chess Congress with interest this year, partly because I have been reading the lovely old book Battles of Hastings: A History of the Hastings International Chess Congress by Reg Cload and Raymond Keene (Pergamon 1991) which contains a nice collection of annotated games from the many years of the Hastings tournament. Hastings must be one of the longest running chess tournaments, and now that the 88th Hastings Masters has itself become history (with the games completed and Gawain Jones entering the annals as its champion), it seemed like as good an excuse as any to use it as a vehicle for exploring chess history. Among my favorite games from the Cload and Keen book are Steinitz - von Bardeleben 1895 (annotated), Bogoljubov - Alekhine summer 1922, Lilienthal - Capablanca 1934-1935, Littlewood - Botvinnik 1961-1962, Kaplan - Bronstein 1975-1976, Keene - Miles 1975-1976 (annotated), and Mestel - Gufeld 1986-1987. But those games just scratch the surface. For a more comprehensive view, check out the "Hastings Christmas Congress (Tournament Index)" that Phony Benoni put together at Chessgames.com indexing all of the great work that he and others have done to collect the games from the historic tournaments.
The great 1895 tournament (won by Pillsbury) is often thought of as the original Hastings tournament. You can read the book of the 1895 tournament edited by Horace F. Cheshire from Google Books (or download it from Chessville), and you can play over the games at Chessgames.com. However, it was not until the winter of 1920/1921 that the tournament became the Christmas Chess Congress that we know today, and the "Summer Congress" events (1895, 1919, 1922, and 1995) are kept separate from the "Christmas Congress" events which now number 88. You can find a complete list of past winners of the premiere event at the old Hastings tournament website (whose history sections have not yet been moved to the new site).