Since its foundation in 1975 it has been one of the most famous chamber ensembles in Slovakia. The Quartet members are graduates of the Academy of Music and Performing Arts in Bratislava. Their studies continued at the University of Music and Performing Arts in Vienna. During their studies the Quartet was already awarded several laureate titles in domestic as well as international contests (Evian). In 1998 they were awarded the Prize of Brazilian Composers for the best CD of Brazilian chamber music. In the years 1986–2005 the Moyzes Quartet belonged to the ensembles of the Slovak Philharmonic.
They successfully represented Slovak music culture in almost all European countries, Japan, India, Morocco, the USA, Canada, the Faroe Islands and Cuba. In 2000 the Moyzes Quartet was awarded the Frico Kafenda Prize and at the same time the Critics Prize. In 2015 the ensemble won the prestigious Tatra Bank Foundation Prize for Arts. In 2016 they were awarded the Slovak Minister of Culture Prize.
The Quartet repertoire encompasses all style periods including works of contemporary Slovak composers, many of whom directly dedicated their pieces to the Quartet. The Quartet has been constantly recording new compact discs — more than 40 CDs already recorded — and making recordings for Slovak radio and foreign radio companies.
During its existence, it gave concerts in important music halls of the world: Schauspielhaus – Berlin, Mozarteum – Salzburg, Gewandhaus – Leipzig, Teatro Real – Madrid, Cathédrale Saint-Louis des Invalides – Paris, Merkin Hall – New York, Walter Hall – Toronto, Musikverein – Vienna, Auditorium – Rome, Small Hall of the Shostakovich Philharmonic – St. Petersburg. Concerts were also held in Barcelona, Brussels, Budapest, Kyiv, Milan, Munich, Moscow, Osaka, Prague, Tokyo, Warsaw, Geneva and many other cities around the world.
For 40 years first violin of the Moyzes Quartet was held by Stanislav Mucha, since 2016 replaced by Jozef Horváth. In 2022 František Török was replaced as second violin by Július Horváth. The viola and cello positions are held by long-term members Alexander Lakatoš and founding member Ján Slávik.