Szathmáry studied composition with Ferenc Szabó and organ with Ferenc Gergely at the Franz Liszt Music Academy in Budapest from 1958 to 1963. He pursued post-graduate instrumental education at first in Vienna with Alois Forer and—after he moved to Germany–from 1964 at the Frankfurt Musikhochschule with Helmut Walcha. Parallel to this he participated from 1964 to 1967 in the Cologne Courses for New Music, studying composition with Henri Pousseur and Karlheinz Stockhausen, and attended the Darmstädter Internationale Ferienkurse für Neue Musik in 1964 and 1965, studying with György Ligeti.
After sitting his A-exams in church music in 1970, Szathmáry worked at first as organist in Hamburg-Wellingsbüttel and from 1976 to 1978 at the Bremen Cathedral. From 1972 he was also active as a lecturer at the conservatories in Lübeck, Bremen, and Hannover; in 1978 he accepted the position of professor of organ at the Staatliche Hochschule für Musik in Freiburg im Breisgau. Besides guest professorships in Tokyo and Seoul, and organ courses (inter. al., as lecturer at the Summer Academy for Organists in Haarlem, the Darmstädter Internationale Ferienkurse für Neue Musik, as well as at numerous conservatories and universities in Europe, North America, Japan, and Korea), Szathmáry has developed a world-wide career not only as an organist and pianist, but also as a conductor.
His artistic activities have been rewarded with numerous prizes and honors: in 1960 he won first prize in the Budapest Organ Competition, in 1972 he was awarded the Bach-Prize Stipend from the city of Hamburg, in 1973 was inducted into the Freie Akademie der Künste in Hamburg, and in 1987 received the Franz Liszt Badge of the Hungarian Liszt Memorial Committee (A Magyar Köztársaság Liszt Ferenc Emlékbizottsága). Since 2007 Szathmáry has been titulary organist at St. Peter’s in Cologne.
As a performer, Szathmáry’s repertoire encompasses organ music from the 17th century up to the present. A differentiated articulation and ornamentation practice as well as a discreet agogic and fluent tempos show him to be an historically oriented, but a thoroughly individual interpreter in the saping of details. Open to musical experiment and technical innovations, in particular within the field of new music, he has earned a considerable reputation: In close co-operation with composers like Péter Eötvös, Vinko Globokar, Heinz Holliger, György Ligeti, Wolfgang Rihm, Peter Ruzicka, Dieter Schnebel, and Hans Zender, he has to date performed about 120 premières and enthusiastically advocates avant-garde organ music (amongst others: Luciano Berio, John Cage, Roman Haubenstock-Ramati, Maki Ishii, Mauricio Kagel, György Kurtág, Giacinto Scelsi).
As a composer Szathmáry pursues an undogmatic pluralism in the application of contemporary compositional procedures, putting a special emphasis on making instrumental tone colors unfamiliar through the use of unusual playing techniques as well as live-electronic and electroacoustic means.