Born in Camprodon, Catalonia, Albéniz was a child prodigy who first performed at the age of four. At age seven he passed the entrance examination for piano at the Paris Conservatoire, but he was refused admission because he took out a ball from his pocket and broke a glass window while playing with it. By age fifteen, he had already given concerts worldwide. After a short stay at the Leipzig Conservatory, in 1876 he went to study in Brussels. In 1880, he went to Budapest to study with Franz Liszt, only to find out that Liszt was in Weimar, Germany.
In 1883 he met the teacher and composer Felip Pedrell, who inspired him to write Spanish music such as the Suite Española, Op. 47. The fifth movement of that suite, called Asturias (Leyenda) is probably most famous these days in the classical guitar world, even though it was originally composed for piano and only later transcribed to guitar by Francisco Tárrega. Many of his other compositions were also later transcribed to guitar - Albéniz himself preferred Tárrega's guitar transcriptions to his original piano works.
During the 1890s Albéniz lived in London and Paris and wrote mainly theatrical works. In 1900 he started to suffer from kidney disease and returned to the writing of piano music. Between 1905 and 1909 he composed his most famous work, Iberia (1908), a suite of twelve piano "impressions".
His orchestral works include Spanish Rhapsody (1887) and Catalonia (1899).
Albéniz died in 1909 at age 48 in Cambo-les-Bains and is buried in the Cementiri del Sudoest, Barcelona.
Cécilia Sarkozy, the (second) wife of French politician Nicolas Sarkozy, is the great-granddaughter of Isaac Albéniz.