Michelle Yeoh deliberately did not work for a year before filming began so she could concentrate on training and learning Mandarin.
The actors almost invariably performed their own stunts. VFX were used to remove the wires holding them up.
"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.
The first draft of the screenplay said, "You will note in the script that none of the fight scenes are described, and I will just inform you now that they will be the greatest fight scenes ever in the history of cinema, period."