Born in Camprodon, Catalonia (Spain) 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 the time he had reached 12, he had made many attempts to run away from home. 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 Felipe 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 as part of the classical guitar repertoire, even though it was originally composed for piano and only later transcribed to guitar. Many of his other compositions were also transcribed to guitar, notably by Francisco Tárrega — Albéniz once declared that he preferred Tárrega's guitar transcriptions to his original piano works.

In 1883, the composer married his student Rosina Jordana. They had three children, Blanca (who died in 1886), Laura (a painter), and Alfonso (who played for Real Madrid in the early 1900s before embarking on a career as a diplomat).

Albéniz died on 18th May 1909 at age 48 in Cambo-les-Bains and is buried in the Cementiri del Sudoest, Barcelona.

Cécilia Sarkozy, the former wife of French president Nicolas Sarkozy, is the great-granddaughter of Isaac Albéniz.