Hadjidakis was born in the northern town of Xanthi, Greece, on October 23, 1925. He started piano lessons at age four, and later learned the violin and accordion as well. In 1932, his parents divorced, and he moved with his mother to Athens. His father died in a plane crash in 1938, leaving the family in a dire financial state only worsened by the German occupation in World War II. Hadjidakis worked a succession of odd jobs to support his family, but also managed to study advanced music theory and composition as a teenager, also enrolling at the University of Athens to study philosophy (circumstances prevented him from finishing his degree). In 1943, he met the revered surrealist poet Nikos Gatsos, who would go on to become his favorite lyricist and work with him on the vast majority of his vocal compositions.
Hadjidakis first found an outlet for his compositional ability when he connected with the Art Theatre of Athens, and contributed music to its 1944 production of Alexis Solomos' The Last White Crow. He would work with the Art Theatre for the next 15 years, scoring a number of canonical plays by American and European writers, and also began to do the same for the Greek National Theatre starting in the early '50s. He wrote his first film score, for Free Slaves, in 1946, and the following year published his first contemporary piano piece, For a Small White Seashell. In 1948, Hadjidakis gave a high-profile academic lecture praising rembetika (sometimes spelled rebetico), the popular folk song form that was the province of the urban lower class and was regarded as borderline immoral by the conservative intelligentsia. The country's musical establishment was scandalized, but Hadjidakis had mapped out the path that would make him one of modern Greece's most cherished musical figures.
As a composer, Hadjidakis embraced rembetika on his 1951 piano work Six Folklore Paintings, which adapted rembetika melodies into a more artful presentation, and was also presented as a ballet (one of four he composed from 1949-1957). In the meantime, he had begun to compose music for theatrical productions of classic Greek tragedies, starting with Aeschylus' Orestes trilogy in 1950; normally, such a job was reserved for scholarly academics. He completed one of his major modern classical works in 1954's The C.N.S. Cycle, a song cycle for piano and baritone vocalist. The following year, he scored the motion picture Stella, which would prove to be one of his major successes in that area. Starring actress Melina Mercouri, whom Hadjidakis had known from her days in the theater, sang the song portions of the soundtrack, and would become one of Hadjidakis' most sympathetic interpreters.
In 1959, Hadjidakis began working with the young, up-and-coming Nana Mouskouri, for whom he would supply material on a regular basis; he also helped introduce the music of Mikis Theodorakis to the Greek public by arranging his song "Epitaphios" for a Mouskouri recording session. The following year, he reunited with Mercouri on the Jules Dassin-directed film Never on Sunday. It was a breakthrough international hit that won Hadjidakis an Oscar for his title song, which became a smash success in many parts of the world. In 1962, he staged the controversial musical Street of Dreams, now regarded as a landmark of Greek theater for its frank subject matter, and completed revisions on his score for Aristophanes' Birds, which subsequently ranked among his finest compositions.
Hadjidakis scored two more internationally prominent films in Elia Kazan's America, America (1963) and Jules Dassin's Topkapi (1964), and struck up a lengthy partnership with choreographer Maurice Béjart of 20th Century Ballets, who collaborated on the composer's forays into ballet from then on. Hadjidakis also founded the Athens Experimental Orchestra in 1964, which provided a vehicle for his own work and that of avant-garde Greek composers like Iannis Xenakis. However, even as Hadjidakis' interest in experimental music grew, so did his interest in song structure; his 1965 work Mythology found him ranging farther and farther afield in his traditional songwriting, fusing elements of symphonic, Turkish Byzantine, and ancient Greek music with modern rembetika.
In 1966, Hadjidakis traveled to New York for the Broadway premiere of Illya Darling, the stage version of Never on Sunday. He wound up staying there until 1972, in part because of the repressive military junta that took over the Greek government. While in America, he completed several more major compositions, including the piano piece Rhythmology and the song cycle Magnus Eroticus, which set 12 Greek poems modern and ancient to music; still fascinated by popular song, he also recorded the LP Reflections with the New York Rock and Roll Ensemble.
Hadjidakis returned to Greece in 1972, and when the military dictatorship fell, he took a number of high-ranking cultural positions: directing the State Orchestra (through 1981) and the classical-oriented channel of the national radio (1982), as well as becoming deputy director of the national opera (until 1977). He started several music festivals and competitions in the late '70s and early '80s, and in 1985 started his own record company (Sirius) and cultural magazine. In 1989, he founded and directed the Orchestra of Colours, a symphonic group devoted to unconventional works. By this time, he was suffering from heart problems, which would eventually claim his life on June 15, 1994.