Since its foundation at the Bratislava Conservatory in 1975 (A. Vrteľ) it has been one of the most important chamber ensembles in Slovakia. The members of the quartet are graduates of the Academy of Performing Arts in Bratislava (T. Gašparek). They continued their studies at the Hochschule für Musik und darstellende Kunst in Vienna (F. Samohyl, G. Pichler). Already during their studies the quartet won several prizes at national and international competitions (Evian). In 1998 it won the Brazilian Composers' Prize for the best CD of Brazilian chamber music (G. Bauer, H. Crowl). From 1986 to 2005 the Moyzes Quartet was one of the artistic ensembles of the Slovak Philharmonic.

The Moyzes Quartet has successfully represented Slovak musical 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 Fritz Kafenda Prize and the Critics' Prize. In 2015, the ensemble received the prestigious Tatra banka Prize for the Arts and was honoured with the Slovak Minister of Culture Award for the Arts. The ensemble's repertoire encompasses all stylistic periods, not excluding the works of contemporary Slovak composers, many of whom have directly dedicated compositions to the quartet.

Quartet is constantly engaged in recording new compact discs (the ensemble has recorded over 40 CDs: A. Albrecht, J. L. Bella, J. Cikker, A. Dvořák, A. T. Grečaninov, E. H. Grieg, L. Kupkovič, W. A. Mozart, A. Moyzes, M. Moyzes, M. Ravel, F. Schmidt, B. Smetana, E. Suchoň, D. Shostakovich, G. Verdi, I. Zeljenka, ...) and making recordings for Slovak radio and foreign radio companies.

During its existence it has given concerts in major 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, Shostakovich Philharmonic Small Hall - St. Concerts have also been given in Barcelona, Brussels, Budapest, Kiev, Milan, Munich, Moscow, Osaka, Prague, Tokyo, Warsaw, Geneva, etc.

The Moyzes Quartet has also recorded music for several successful Slovak films. Since 2006 the Moyzes Quartet has been the chamber ensemble of the town of Modra. In 2008 it became the chamber ensemble of the town of Skalica. For 40 years, the Moyzes Quartet was directed by Stanislav Mucha, and since 2016 it has been directed by Jozef Horváth. In 2022, František Török was replaced as second violin by Július Horváth (a member of the 1st violin group of the Slovak Philharmonic). Alexander Lakatoš (member of the Slovak Radio Symphony Orchestra) and Ján Slávik (founding member of the Moyzes Quartet and concertmaster of the Slovak Philharmonic Cello Group) are long-standing members of the viola and cello positions.

This year the Moyzes Quartet celebrates its 49th anniversary. In recent years, the quartet has recorded 2 CDs with three string quartets by J. N. Hummel and a CD with three string quartets by Vladislav Šarišský. In 2021 it recorded its new CD with works by Slovak composers (A. Moyzes, I. Zeljenka, M. Varga, L. Burlas, E. Krák). This year the quartet is recording three masterpieces by Joseph Haydn for the Real Music House label.