Born in Plovdiv, Bulgaria, Yoncheva did not have a conventional childhood. Since the age of 6, she was already "working", according to her own words from an interview for Metropolis. She studied piano and voice at the National School for Music and Dance in Plovdiv. During her teenage years, she hosted a Bulgarian television show about music. She won several music competitions in Bulgaria in 2000 and 2001, including a joint win with her brother as "Singers of the Year 2000" in the "Hit-1" competition organised and produced by Bulgarian National Television.
Yoncheva was an invited participant in William Christie's "Jardin des Voix" academy for young singers in 2007. She has continued work in the baroque repertoire with Christie, and also with Le Concert d'Astrée and Emmanuelle Haïm. Yoncheva was the female first-place winner of the 2010 Operalia competition. Her debut at the Metropolitan Opera was in November 2013 when she stepped in as Gilda in Rigoletto, in advance of her previously scheduled official debut. In 2014 at the Met, she sang her stage role debut as Mimì (La bohème) in November and shared the role of Violetta (La traviata) in December. In 2015 she appeared as Desdemona in the season opening production of Otello the Metropolitan Opera and in September 2016 she made her debut as Norma at the Royal Opera House, where she sang Antonia in Les contes d'Hoffmann in October of the same year. In June 2017 she made her debut at the Teatro alla Scala as Mimì in La bohème.  She made her role debut in the title role of Puccini's Tosca at the Metropolitan Opera for their 2017/18 season to great critical acclaim. Yoncheva set a record as first singer to appear in leading roles in three movie-theater broadcasts of the Metropolitan Opera in one season (Tosca, Mimi in La bohème and Luisa Miller in the 2017/18 season).  Other role debuts in the 2017/18 season include Verdi's Luisa Miller at the Metropolitan Opera, Élisabeth de Valois in Verdi's Don Carlos at the Paris Opera and Imogene in Vincenzo Bellini's Il pirata at the Teatro alla Scala. In summer 2018 Yoncheva will sing the role of Poppea in Claudio Monteverdi's L'incoronazione di Poppea at the Salzburg Festival.
Yoncheva is featured on commercial recordings on the Virgin Classics label. In November 2013, she signed a recording contract with Sony Classical. She also appears in commercial DVD releases of Claudio Monteverdi's L'incoronazione di Poppea and Il ritorno d'Ulisse in patria, and of Giovanni Battista Pergolesi's Il Flaminio. Additionally, she is featured on Pergolesi's "Stabat Mater" and Mozart's "Le Nozze di Figaro" on Deutsche Grammophon.
Her first solo album, the audio CD Paris mon amour (Sony Classical) came out on January 13, 2015, followed by another solo album titled Handel (Sony Classical) released on February 3, 2017 and the solo CD The Verdi Album, released on February 2, 2018.
Yoncheva and her husband Domingo Hindoyan, a Venezuelan conductor, live in the canton of Vaud, Switzerland. The couple have a son, Mateo, born 6 October 2014. Her younger brother Marin Yonchev is a former rock singer and is now appearing in operatic concerts alongside Yoncheva.
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