Per Rundberg was born in Skellefteå, Sweden, and studied with Björn Ejdemo, Staffan Scheja, and Gunnar Hallhagen in his native country before attending the Yehudi Menuhin School, Stoke d´Abernon, England at the age of 15, where he continued his studies with Seta Tanyel. Later studies in Salzburg at the Mozarteum (with Karl-Heinz Kämmerling, resulting in a diploma with honors), Paris, Budapest, and Stockholm, as well as lessons with Murray Perahia, completed his education.

Per Rundberg frequently gives concerts all over Europe, in the USA and in Asia. He has performed in major halls including the Concertgebouw (Amsterdam), Konzerthaus and Musikverein (Vienna), Philharmonie and Konzerthaus (Berlin), Carnegie Hall (New York), Konserthuset and Berwaldhallen (Stockholm), Teatro Alla Scala, (Milan) and Philharmonie (Cologne) and as a soloist with such orchestras as the Swedish Radio Symphony Orchestra, The Royal Opera Orchestra (Stockholm), Camerata Salzburg, Mozarteum Orchester Salzburg, Morávska Filharmonie (Czech Republic), and Ostrobothnian Chamber Orchestra (Finland), among others. He is often invited to play at major festivals like Schleswig-Holstein, the Salzburg Festival, Beethovenfest, Bonn Heidelberger Frühling, and Festival di Pasqua in Rome. Among his chamber music partners are and have been Dietrich Fischer Dieskau, Martin Grubinger, Thomas Hampson, Valentin Răduțiu, and Benjamin Schmid.

Together with Martin Grubinger, he was selected by the Wiener Konzerthaus for a Rising Stars -tour around the world in 2008.  Per Rundberg has recorded CDs and DVDs on various labels like Accent Music, Deutsche Grammophon, Hänssler Classics, Oehms, and KC Classics. He performed the complete piano works by Wolfgang Amadeus Mozart in Salzburg over several years.

Dedicated to contemporary music, Per Rundberg has given many world premiere performances; he has worked with some of the most prominent composers of our time, including George Crumb, Franco Donatoni, Beat Furrer, Giya Kancheli, György Kurtag, and Arvo Pärt.