His debut as a concert pianist was in 1861 in Karlshamnm, Sweden. After he finished his studies in Leipzig in 1862, Grieg held his first concert in his home town, where his programme included Beethoven's Pathétique sonata. After a 3 year stay in Copenhagen (1863-1866), where he met the young Norwegian nationalist composer Rikard Nordraak, he got acquainted with the northern folk tunes and had learned more about his own nature. After Nordraak died in 1866, Grieg composed a funeral march in the honor of his good friend.
In the winter of 1864–65 Grieg became one of the founders of the Copenhagen concert society, Euterpe, for the production of works by young Scandinavian composers. On 11 June 1867, he married his first cousin, Nina Hagerup, a lyric soprano who became an authoritative interpreter of his songs. In the summer of 1868, while on holiday in Denmark, Edvard Grieg wrote his Piano Concerto in A minor. Edmund Neupert gave the concerto its premiere performance on 3 April 1869 in the Casino Theater in Copenhagen. Grieg himself was unable to be there due to conducting commitments in Christiania (as Oslo was then named). The piece was received with an enthusiasm that would attach itself to the composer's reputation for the remainder of his career.
Having spent his winter in Rome (1869–70), he met Liszt, who was roused to enthusiasm by his piano concerto. In 1874–76, Grieg composed incidental music for the premiere of Henrik Ibsen's play Peer Gynt, at the request of the author. The premiere was performed to critical acclaim and eventually led to Grieg's scoring of Peer Gynt into Suites 1 and 2 (1888 and 1893 respectively). As a result of the success of Peer Gynt, Grieg enjoyed tremendous celebrity and continued to travel extensively, often meeting internationally renowned composers such as Tchaikovsky, Brahms, and Liszt, among others.
In the spring of 1903, Grieg recorded some of his piano music in Paris, all these historic discs have been reissued on both LPs and CDs, although, with limited fidelity. In 1906, he met the composer and pianist Percy Grainger in London. A strong friendship was quickly established as Grainger was a great admirer of Grieg's music. On top of that, Grieg admired Grainger for his unique and accurate interpretation of his (Grieg's) songs. In a 1907 interview, Grieg stated: “I have written Norwegian Peasant Dances that no one in my country can play, and here comes this Australian who plays them as they ought to be played! He is a genius that we Scandinavians cannot do other than love.”
Some of Grieg’s early works include a symphony (which he later suppressed) and a piano sonata. He also wrote three violin sonatas and a cello sonata. Rooted in the national folk tradition of Norway, Grieg’s music is noted for a refined lyrical sense. Between 1867 and 1901 he wrote ten collections of Lyric Pieces (Lyriske Stykker) for piano.
His music incorporated clear signs of spirited rhythms that have a folk song association; his harmonies, rooted in the late Romantic style, were considered novel. In his few works in the larger forms—the Piano Concerto, Opus 16; the String Quartet in G Minor, Opus 27; and the three violin and piano sonatas—he uses a free sonata form. His original Ballad for piano, Opus 24, is a set of variations on a folk theme. Among his most popular works are his incidental music to Peer Gynt, Opus 23, and the suite Holberg, Opus 40. His arrangements of Norwegian dances and songs, Opus 17 and Opus 66, and especially his Slåtter, Norwegian Peasant Dances, Opus 72, show his characteristic sense of rhythm and harmony. His vocal works include the songs on texts of A.O. Vinje, Opus 33; and the Haugtussa cycle, Opus 67.
His use and development of Norwegian folk music in his own compositions put the music of Norway in the international spectrum, as well as helping to develop a national identity, much as Jean Sibelius and Antonín Dvorák did in Finland and Bohemia, respectively. Grieg is regarded as simultaneously nationalistic and cosmopolitan in his orientation, for although born in Bergen and even buried there, he traveled widely throughout Europe, and considered his music to express both the beauty of Norwegian rural life and the culture of Europe as a whole.
Here you can find a list of Edvard Grieg's compositions.