Born in Tel Aviv, Israel, Shmuel Ashkenasi attended the Musical Academy of Tel Aviv and gave his first public performance at the age of eight. After studying with Ilona Feher, he came to the United States to study with Efrem Zimbalist at the Curtis Institute of Music. He won the Merriweather Post Competition, was a finalist in the Queen Elisabeth Competition, and received second prize in the International Tchaikovsky Competition in Moscow.

Shmuel Ashkenasi has toured the former Soviet Union twice and concertized extensively in Europe, Israel, the Far East, and the United States, and he has collaborated with Rudolf Serkin, Thomas Hampson, Murray Perahia, Peter Serkin, and Menahem Pressler. As first violin of the famed Vermeer Quartet, he has gained a reputation as one of the world's outstanding chamber musicians.

From 1969 until 2007, Shmuel Ashkenasi was professor of music and artist-in-residence at Northern Illinois University, and, for the last several years, he taught at Roosevelt University in Chicago.
He joined the faculty of the Curtis Institute of Music in 2007.