Yuri Matveyevich Felten was born in 1730. His father, Matthias Felten, arrived in Russia from Germany and was appointed an economist at the newly established St. Petersburg Academy of Sciences in 1725. He named his son Georg Friedrich, but later his name was Russified to Yuri Matveyevich. In 1736, his father died, and the care of the family was taken over by Yuri's sister's husband. Together with him, the Felten family moved to Germany. Before leaving for Tübingen, Yuri graduated from the senior classes of the gymnasium at the Academy of Sciences, which allowed him to enroll at Tübingen University.

While studying at the university, Felten traveled around the country, studying architecture. He interned during the construction of the New Palace of the Duke of Württemberg in Stuttgart in 1747-1748, and in the summer of 1749, in Berlin.

In the fall of 1749, Felten returned to St. Petersburg and in November submitted a request to Empress Elizabeth Petrovna to enroll in the Academy of Arts. In December, his request was granted, and he became the first student-architect along with Reinhard Ash. Architecture was taught by I. Y. Schumacher, who not only provided theoretical knowledge but also engaged students in practical activities. In November 1751, Felten submitted a request to join the Academy's service.

On December 20, 1755, by a decree of the Senate, Felten was accepted into the service of the Office of Buildings and appointed assistant architect, starting to work in the team of Francesco Bartolomeo Rastrelli. Initially, he did not receive a permanent position, but later he was sworn in and in 1760 was accepted as an architect with Rastrelli's consent, whom he assisted in the construction of the Winter Palace.

After Rastrelli's retirement, Felten, along with Vallin-Delamot, became one of the leading architects in St. Petersburg. He completed Rastrelli's projects, including work on the Winter Palace and the adaptation of the Smolny Monastery buildings for the Institute of Noble Maidens. He was entrusted with the construction of a school for merchant girls.

In 1762, Felten was appointed chief architect of the Office of House and Garden Construction, succeeding Rastrelli. This position allowed him to oversee significant projects in St. Petersburg, including the construction of the stone Palace Embankment and its fence, as well as the buildings of the Great Hermitage and the arch over the Winter Canal.

In 1765, Felten participated in a competition to restore the bell tower of St. Peter and Paul Cathedral, after which he focused on building church structures. He built the Swedish Church of St. Catherine, the Armenian Church on Nevsky Prospect, the Lutheran Church of St. Catherine on Vasilievsky Island, the Church of John the Baptist, and others.

The Chesme Palace and Church, as well as the Church of John the Baptist, became unique examples of Gothic architecture in the suburbs of St. Petersburg.

In 1773, Felten reconstructed the house of Captain Glazyev in the square near the Winter Palace, which allowed him to win the competition for the reconstruction of the entire palace space. According to his project, buildings were constructed on the southern border of the future Palace Square.

Empress Catherine II appointed Felten responsible for the installation of the monument to Peter I (The Bronze Horseman). He became the assistant to the sculpture's author, Étienne Maurice Falconet, and after his departure, he oversaw the installation of the monument.

Felten lived in a mansion on the Moika River, where he built a stone house. However, he did not find family happiness: in 1774, his wife Anna and their children died.

From 1776 to 1782, he built a palace on Kamenniy Island for Paul, the son of Catherine II. Felten also worked in Peterhof and Tsarskoye Selo, creating the interiors of the Grand Peterhof Palace.

In the early 1780s, Felten completed the construction of the Academy of Arts building, after which he moved into a service apartment and sold his mansion.

After Vallin-Delamot's departure from Russia, Felten began teaching architecture at the Academy of Arts, receiving the title of professor after the death of A. F. Kokorinov. Since 1775, he had been the senior professor, fully immersing himself in the affairs of the Academy.

For his long service, Felten was awarded the rank of state councilor. In 1783, he was elected a correspondent of the French Royal Academy.

In 1785, Felten became the adjunct rector of the Academy, and in 1789, its director. In 1794, he left the Academy without any awards, and his request was denied. The architect died on June 2, 1801. Among his notable students were Andreyan Zakharov and the brothers Alexander and Andrey Mikhaylov.