I have rigged my character with automatic weights, and everything seems OK except for a few faces which are left behind when I rotate the head. They are the faces that make up the iris and pupil in the eyes.
These faces were modelled separately to the rest of the eyes but they were joined to the eye objects so I don't know why they don't follow them! I'm not able to select them as separate objects, yet rotating the bones in pose mode leaves them behind as you can see in the image below.