Allevi attained a "first-class diploma" both in piano at the "F. Morlacchi"'s conservatoire in Perugia and in composition at the "G. Verdi" academy of music in Milan. He graduated cum laude in Philosophy with a thesis entitled "Il vuoto nella Fisica contemporanea" (The Void in Contemporary Physics) and he attended the Accademia Internazionale di Alto Perfezionamento in Arezzo, under maestro Carlo Alberto Neri.

In 1991 he did military service in the Banda Nazionale of the Italian Army. The master of the band noticed his piano talent and decided to put the piano soloist in his "inventory." He played the Rhapsody in Blue by George Gershwin and the Warsaw Concerto by Richard Addinsell in his capacity as piano soloist of the Banda during a tour in numerous Italian theatres. At the end of his military service Allevi started presenting in concert a repertory formed exclusively of his own compositions for pianoforte while at the same time attending courses of "Bio-music and music therapy" by professor Mario Corradini, in which he enhanced his awareness of music's great power of "setting minds free" and its ability to evoke memories, images and emotions.

In 1996 he set to music the tragedy The Trojan Women by Euripides, performed on the occasion of the International Festival of the Ancient Drama in Siracusa. It won a special prize for the "best scene musics."

In 1997 he won the international selections for young concertists at the San Filippo Theatre in Turin.