Alessandrini was born in Rome on January 25, 1960. He was relatively late in beginning his musical studies, taking up the piano at age 14. Four years later, he discovered the harpsichord and traveled to the Netherlands for lessons with Ton Koopman at the Sweelinck Conservatory in Amsterdam. He was soon giving concerts on the harpsichord. Alessandrini founded Concerto Italiano in 1984. He also played the organ, and his first recording, a performance of Girolamo Frescobaldi's Fiori Musicali collection, was on that instrument. That recording was on the Naïve label, which would be Alessandrini's home for much of his recording career. In 1992, he issued a collection of Alessandro Scarlatti's keyboard music on the Arcana label; he has also recorded for Opus 111, which released Alessandrini's early recordings of Monteverdi madrigals with Concerto Italiano.
As an opera conductor, Alessandrini has led performances of Monteverdi operas as well as those of lesser-known composers, such as Francesco Cavalli and Leonardo Vinci. His involvement with a major series of recently rediscovered Vivaldi works issued on Naïve brought him to international attention and resulted in Europe-wide tours and festival appearances for Concerto Italiano. Other acclaimed Alessandrini recordings included those of Handel's Italian-language operas and harpsichord concertos by J.S. Bach, stressing their direct Italian antecedents. The group's recordings of Bach's Brandenburg Concertos and Vivaldi's Four Seasons violin concertos have been both critical and commercial successes. Alessandrini returned to Monteverdi's operas in the 2010s, conducting L'incoronazione di Poppea at the La Scala opera house in Milan. He has recorded prolifically, issuing as many as four albums a year in the 2010s; the 2018 set included Un viaggio a Roma.
Alessandrini is a Chevalier of the Order of Arts and Letters, by the designation of the French government. ~ James Manheim, Rovi