Anthony Quinn had a broken foot during filming, and thus couldn't perform the dance on the beach as scripted, which called for much leaping around. The dance is called 'syrtaki' and is very traditional.
Simone Signoret was offered the part of Madame Hortense initially, and turned it down. She came to regret that decision. The role went to Lila Kedrova who ended up receiving an Academy Award for her role.
The original Broadway production of "Zorba" opened at the Imperial Theater in New York on November 16, 1968, starring Herschel Bernardi and Maria Karnilova. It ran for 305 performances and was nominated for the 1969 Tony Award for the Best Musical. Anthony Quinn and Lila Kedrova, who starred together in this movie version in 1964, appeared in the revival of Zorba in 1982/83. Lila Kedrova won the 1984 Tony Award for Best Featured Actress in a Musical.