The actors almost invariably performed their own stunts. CGI was used to remove the wires holding them up.
Michelle Yeoh deliberately did not work for a year before filming began so she could concentrate on training and learning Mandarin.
"Crouching tiger hidden dragon" is a quote from Chinese mythology. It refers to hiding your strength from others.
Michelle Yeoh tore her ACL (anterior cruciate ligament) during the shooting of an early fighting sequence and had to be flown to the United States for knee surgery. She returned to the set at different times to film non-action scenes until her knee had recovered.
While Ziyi Zhang's character is highly trained and skillful in martial arts, the actress herself never had any official martial arts training. Instead, she used her dance techniques to learn her moves as if they were a dance rather than a fight (which, in terms of creating and filming them, is basically the same).