In reply to Catriona:
A standard handlebar grip will fit, as they age they become looser and start to slip and this can sometimes polish the bar. Generally speaking new grips will stick, to make it easier you can spray some hair spray on the inside of the grips, this will help the grip to slide on the bar and when the hairspray evaporates the grip will stick, if they still slip which is generally on wet days you sometimes need to rough the bar with grit paper to give some more friction. I use lock on grips with small Allen keys on both ends so I don't have this problem.