The first composers to write polyphonic settings of the text were Guillaume DuFay and Johannes Ockeghem but sadly, Ockeghem's is the only one to survive. Here is the Introit:
There are over 2000 settings of the Requiem, some intended for use within a liturgical setting, others as concert music. Here is a rather passionate a capella setting by Tomás Luis de Victoria dating from 1603:
One of the most famous settings of all is that of Mozart, unfinished as he lay on his deathbed in 1791:
As you can hear, by this time the settings were usually for vocal soloists, choir and full orchestra.
One of the most grandiose and powerful settings is that of Hector Berlioz in 1837 which he titled Grande Messe des Morts. Apart from the usual forces mentioned above, the instrumentation includes four brass bands placed in four corners of the hall and ten, TEN, tympani players with a lot of other percussion as well. I have had the experience of hearing a live performance and there are moments when you are almost sure that the actual Apocalypse itself could not be nearly so impressive! Here is a very grand performance conducted by Colin Davis. The actual performance starts at the 13 minute mark:
You can see where Mahler got some of his ideas. Now, of course, composers outside the Catholic tradition also did similar kinds of settings. In the case of Brahm's German Requiem of 1868 he uses texts, in German, from the Lutheran Bible:
This was followed in 1874 by Verdi's setting of the traditional Catholic text:
Wikipedia has an astonishingly touching story of performances of this piece in WWII:
The Requiem was performed 16 times between 1943 and 1944 by prisoners in the concentration camp of Theresienstadt (also known as Terezín) under the direction of Rafael Schächter. The performances were extraordinary on several counts: first, they had only a single vocal score with piano accompaniment, so every part had to be learned from memory; second, they practised in a dark, cold, damp basement with only a broken piano after long days of forced labour; and third, as the performances took place over an extended period, many of the singers were removed by the Nazis and had to be replaced. The final performance particularly provided a basis for dignified self-expression as well as attempting to symbolically communicate the circumstances at the camp to a visiting International Red Cross delegation in 1944.Theresienstadt was the concentration camp where the Nazis sent children and artists, including musicians. A very dear friend of mine, violinist Paul Kling, was sent there as a child and later moved to Auschwitz. Unlike many, he survived...
Another requiem, perhaps the most elegant and charming, is by Gabriel Fauré, composed in 1890:
The 20th century saw some unconventional settings such as that by Benjamin Britten in 1962 called the War Requiem. In addition to the traditional text, the setting includes nine poems by the war poet Wilfred Owen. Here is a performance conducted by John Eliot Gardiner:
In 1966 Igor Stravinsky made a brief, partial setting of the text titled Requiem Canticles. The piece, in nine short movements, uses serial technique:
Unless Philip Glass starts writing requiems, we may have come to the end of this history, with one final mention. In the 1980s, in support of the movements for Polish independence, Krzysztof Penderecki made a setting called Polish Requiem. The text is from the traditional liturgy with some additions. There is not a clip on YouTube with the whole piece, but here is the beginning to give you a sense of it:
This five hundred year tradition may have come to an end, but, like the symphony, it may not be over yet.
UPDATE: In a comment on another post, commentator Nathaniel Garbutt recommends filling in the gap in my discussion of the requiem between the late Renaissance and Classical eras with requiems by Charpentier and Biber. He particularly recommends a Biber requiem composed in 1692. Turns out there are two Biber requiems, one in F minor and one in A major. Here are excerpts from both:
Both fine works! Thanks, Nathaniel. Makes me think that Heinrich Biber, who was employed by the Archbishop of Salzburg (who later employed Mozart) for the latter half of his career, really deserves more attention. Perhaps I will do a post on him. A friend of mine did her doctoral dissertation on him.