Like all instruments the clarinet comes alive when you play it. In general what you play will be composed music - only a small part of what clarinet players play is improvised like in Jazz or some folk music like Klezmer. Some compositions are important, famous milestones for clarinet music and therefore well known even to a wide public like Mozart's and Weber's clarinet concertos, the beginnning of Gershwin's Rhapsody in Blue or "the cat" in "Peter and the Wolf".
In the following you find a short list of the most famous and important composers from a clarinet perspective. These are no full CVs, just micro-summaries. There may be links showing more detailed information; in any case you can use a search engine and usually find as much text as you like.
Off course the selection here is subjective. You won't find Johann Sebastian Bach, Beethoven and Wagner: In Bach's days there were no clarinets, and as far as I know neither Beethoven nor Wagner composed something so special or spectacular for us clarinet players (though they were masters at using the clarinet as a symphonic instrument). If you think someone is missing here, do mail a suggestion with an explanation, why you would think the composer should be on the list: firstname.lastname@example.org
before the classical period (Baroque):
Georg Friedrich Händel (1685 - 1759),
Carl Philipp Emanuel Bach (1714 - 1788)
In late Baroque, that is until about 1750, the clarinet could not be seriously used on the on hand, and on the other there were no clarinet players except for a few travelling virtuosos. As result the known composers did not write for clarinet. But already C.P.E. Bach and Händel wrote smaller parts for the new instrument.
Johann Stamitz (1717-1757)
Stamitz not only was an excellent clarinet player, but at the same time the conductor of the Mannheim royal orchestra ("Mannheimer Hofkapelle") and wrote music for that ensemble. The Mannheimers were not necessarily the very first orchestra that had clarinet players as fixed part of their wood wind section (starting 1757), but they added considerably to the clarinet's popularity. This was mainly due to the quality of the orchestra itself and its concert tours going all over Europe. Stamitz wrote a number of pleasant solo concertos that influenced all later composers, including, off course, Mozart.
Wolfgang Amadeus Mozart (1756 - 1791)
The most famous composer has written the best known classical piece of music for clarinet, too: The concerto in A (its slow second movement is even better known to some people as film music from "out of Afrika". Mozart lived in a time when the clarinet still was new and therefore interesting on the one hand, but technically advanced that these difficult pieces could already be mastered. In the year 1778 Mozart wrote to his father: "if only we had clarinets! You can't imagine what a grand effect a symphony with flutes, oboes and clarinets make!" From 1780 on the clarinet was Mozart's favorite instrument and later especially the bassett horn (a deep clarinet) and he became a close friend with the infamous clarinet player Anton Stadler. For him he wrote the most famous of clarinet concertos.
Jan Václav Knezek (1745 - 1806)
The Boehmian [TODO: Link] composer has written solo concerts that can be heard on radio frequently, they have pleasant tunes and show of the strengths of the instruments, especially in the concerto for two clarinets. Knezek worked in the court's orchestra of Thurn und Taxis as an employed clarinet player and composer. That was a good job, and he was free and earned extra premiums for compositions - much in contrast to his friend, the well known french horn player Stich, alias "Punto" (punto means sting - German: Stich - in Italian), who was a musical slave in the same court. Both fled to Germany - where slavery was abolished much earlier - later.
František Kramár (1759 - 1831) [alias Franz Krommer]
The Boehmian [TODO: Link] composer has written several nice solo concertos, that are played frequently on classical radio stations. They are artisitic, but pleasant to hear. Kramár's style is typical for his time which was the time of travelling clarinet virtuosos. They ordered impressive works by famous composers quite often because the audience wanted to hear something new and spectacular.
Louis Spohr (1784 - 1859)
Today Louis Spohr is not as well known as Mozart or Weber, but in his days he was very popular. He started to write for his friend, the clarinet player Johann Simon Hermstedt (1778-1846). Hermstedt was a technically brilliant player, and all four concertos (1808-1829) are full of technical difficulties in all of the clarinet's registers. In the beginning Spohr did not exactly know what was possible on the clarinet and did not make it easy for Hermstedt, who in return improved the mechanics of his instrument. This gave the technical development of the clarinet quite a push.
Karl M. v. Weber (1786 - 1826)
Next to Mozart Weber's compositions for clarinet are the most famous and most frequently played ones. He wrote them for his friend, the clarinet player Heinrich Baermann. Weber knew strengths and weaknesses of the instrument much better than Spohr - and then Baermann was not so much a technical player as Hermstedt. This makes Weber's pieces so satisfying. After one successfull concerto he wrote to a friend: "The whole orchestra went crazy and demands concertos from me. They are running wild writing to the King (Ludwig of Bavaria) and the musical board..."
Brahms already had stopped to compose, but very much like Mozart met Stadler and wrote for him as Weber wrote for Baermann, Brahms met Mr. Mühlfelds and wrote the clarinet trio op. 114, the quintett op. 115 (1891) and the two clarinet sonatas op. 120 in f major and Eb minor (1894). That was a hundred years later than Mozart, and the Instrument was advanced in technique and sound, which reflects in the result.
George Gershwin (1898 - 1937)
George Gershwin was one of the first writing symphonic jazz; that is using jazz elements in a piece that is composed for symphonical orchestra. He thereby created a tradition within which many symphonic composers who write for musical and movies continued until today like Aaron Copland, Leonard Bernstein and John Williams.
One can certainly discuss in how far George Gershwin is part of the romantic tradition (I do think so) or
should be considered "modern". If you have an opinion why don't you write a mail on this?
Sergej Prokofieff (1891 - 1953)
Russsian modern music - that is what Prokofieff stands for, even if you may not think of this at first. His most famous work, "Peter and the Wolf", is in style and design really "modern". But Prokofieff was convinced that the modern composer must be popular, too, or else the audience would just listen to primitive pop music - and so he wrote music, that will easily fill today's concert halls.
"Peter and the Wolf" is not a clarinet concerto, but in the beginning the musical characters are introduced: Peter (violines), the bird (flute), the duck (oboe), the grandfather (bassoon) and the cat (clarinet). In just one little, harmless sounding tune (but try to play it yourself...) Prokofieff demonstrates the ability and characteristics of the clarinet. Probably the most famous clarinet tune of them all.
Bela Bartok (1881 - 1945)
This composer has his roots in Hungarian folk music, which is an asian music (not to be mixed up with the the like of Johannes Brahms' Hungarian dances - those are mostly gipsy tunes). Bartok's pieces have unusual rhythmic concepts. He wrote some pieces for clarinet that Benny Goodman ordered, and this alone already gave them some public interest.
Francis Poulenc (1899 - 1963)
French modern music - Pulenc was a member of the group of "Six" in Paris. He had a radical approach to modern music, but had more traditional periods, too. Today some of his works are standard pieces for the clarinet student - that is when you are somewhat advanced. Nevertheless this will hardly ever become popular music and will hardly ever fill a concert hall. So you won't hear much of him on commercial radio stations in prime time...
Darius Milhaud (1892 - 1974)
Being one of the French modern composers Milhaud is not widely known outside of the French speaking world. That is a pity since he wrote over 450 pieces, some in really popular style, especially the "Scaramouche" - Latin American dances - he composed for Benny Goodman. One of those dances, the "Braziliera", is played on radio from time to time. You can hear that Milhaud did work for France's diplomatic service in South America for some time. Despite being modern, Milhaud was a pleasant harmonist, who was faszinated by rhythm.
Aaron Copland (1900 - 1990)
Like Gershwin, Bernstein, Milhaud and B. Goodman, Aaron Copland too had his roots in the jewish musical tradition - this shows, too, in his personal history as student of Rubin Goldmark in New York and Nadia Boulanger in Paris. His works as early composer - and then again as very late composer - were uncompromising modern, partly 12-tonal (and as result they were not successful from a commercial point of view). But in between he had a phase in which he wrote popular and even jazz-like style (somewhat like Gershwin). In this phase he wrote his clarinet concerto, being the best known "classical" piece of clarinet music composed inside the USA - again a composition written for Benny Goodman.
Gerald Finzi (1901-1956)
An English composer who has written an attractive, pastoral clarinet concerto.