In 1991 Várjon became the first prizewinner at the Géza Anda Competition in Zurich. Shortly thereafter he gave his début at the Salzburg Festival with the Camerata Salzburg under Sándor Végh. He appears on a regular basis in the musical capitals of Europe and the United States and has been a visiting performer inter alia at the Ruhr Piano Festival, the Lucerne Festival, András Schiff’s "Ittinger Pfingstkonzerte", the Weimar Art Festival, the Rheingau Festival, the Schleswig-Holstein Festival, and the Edinburgh Festival.
Várjon has made solo appearances with such leading ensembles as the American Symphony Orchestra, the Camerata Bern, the Orchestre de Chambre de Lausanne, the Zurich Tonhalle Orchestra, the Hungarian State Orchestra, the Academy of St. Martin-in-the-Fields, the Orchestre National du Capitole de Toulouse, the Scottish Chamber Orchestra, the Berlin Radio Symphony Orchestra, the Budapest Festival Orchestra, and the Kremerata Baltica. Among his partners at the conductor’s rostrum were Adam Fischer, Iván Fischer, Leopold Hager, Horst Stein, and Georg Solti.
Besides his solo performances, Várjon also appears alongside leading musicians in chamber music recitals. He enjoys close associations with the composer-performers Heinz Holliger and Jörg Widmann and has issued a number of recordings for various labels.