Following some of Bashmet's prices:
1976 first prize at the ARD music competition
1986 National Prize of the USSR
1993, 1995, 2000 State Prize of the Russian Federation
1995 Léonie-Sonnig music prize
In 1998, his recording of the 13th String Quartet by Shostakovich and the Brahms Quintet received a Grammy nomination and was awarded the "Best Album 1998" by Strad-Magazine.
For a CD with the violin concerto of Sofia Gubaidulina and Giya Kancheli's "Styx," he was nominated for the Grammy and was awarded the "Diapason d'Or" record price.
In 2008 the „Moscow Soloists“ were awarded the Grammy (best performance of small ensembles).

Medals and Honors
Yuri Bashmet is currently considered to be one of the most prominent musicians and is an absolute leader in the world of viola. The following are just some of the most important honors that Bashmet has received. His numerous honors include the early honorary titles of the RSFSR (1983) and the USSR (1991).
1999 honors of the French Ministry of Culture and the Lithuanian government. In 2000, he received an award from the Italian President, and in 2002 he received the Order of Merit for the Fatherland from Vladimir Putin. He is an honorary citizen of the city of Rousse in Bulgaria, honorary professor of the Moscow State University and he is an honorary member of the London Royal Academy of Music and the French "Légion d'honneur".

Bashmet received more than 50 individual honors from various composers.

So many pieces have been written or dedicated to Yuri Bashmet. Particularly noteworthy are Sofia Gubaidullina's violin concerto, John Taverner's "The Myrrh Bearer," Giya Kancheli's "Styx" and various works by Alfred Schnittke.
In 1989, Bashmet called Schnittke the greatest living composer in the world. He would be the only one to let Shostakovich's spirit live on.

Bashmet and his homeland Russia
As a cultural framing of the Winter Olympiad in Sochi, Bashmet directed the VII International Art Festival. On the excitement of the concerts he said:
 
"In Soviet times, when someone drove to international competitions, he wanted to win on his own, of course, and that demanded a lot of work. But on the other hand, there was the million-strong hinterland, the tremendous responsibility! Even now, I feel this: one does not just have to perform well, the cultural program must sound alone for itself, the whole world will see and hear! We all have to do our best: here is my fatherland and not just my profession."
 
Yuri Bashmet would often have had the opportunity to exchange cold Russia for a chic warm place. Even in the time of the Soviet Union, when numerous artists left the country, and the Russian art scene began to bleed, emigration to Bashmet never came into question. He was always keen to preserve the Russian cultural landscape.

In 1991 he toured with the "Moscow Soloists" in France, where after the concert most artists did not return to Moscow. Bashmet resigned from his position as director of the orchestra. However, he founded the "Moscow Soloists" again after his return with new musicians.

He described Moscow as his cultural center. Munich or Paris are cities, which one gladly visits but also gladly leaves again.
 
In 2014 he was one of the "cultural creators" who signed an open letter to support Putin's position in the Ukraine crisis.
This gesture cost him the honorary professorship at the National Music Academy Lviv (Ukraine). The title was invalidated the same month.

Sources:
https://rbth.com/articles/2009/09/10/100909_bashmet.html
http://violacentral.com/spotlight-yuri-bashmet/
http://www.allmusic.com/artist/yuri-bashmet-mn0000537571
https://en.wikipedia.org/wiki/Yuri_Bashmet
 
Sound examples:
https://www.youtube.com/watch?v=zzKs4JV9-qo
A recording of Hindemiths Sonata for Viola, op.11 Nr. 4; 1985 in Moscow. Yuri Bashmet is playing with Sviatoslav Richter
 
https://www.youtube.com/watch?v=pI0F83eQElU
The third movement of Schnittke's Viola Concerto. The Concerto was dedicated to Bashmet. Yuri Bashmet is playing the solo viola, Waleri Gergijew is conducting.

https://www.youtube.com/watch?v=NfZ0JtMxhW4
Spivakov-Bashmet Mozart Duos