Hopefully the project does not have to start over (maybe the reason why the PM had to leave), in which case the new PM would start from the beginning like every other time – as long as they know the content. If the PM shifted during the middle of the project, I would agree with the “kick off” meeting that was previously mentioned by a few. I think this would be the best way to get to the new PM up to speed. I also think it would be beneficial if the old PM would “show [me] the ropes”. Hopefully the old PM is amicable about leaving and willing to share information and strategies that will help the new PM.
If the PM is changing near the end of the project, I actually think it would be good for the person to stay on for as long as they can (except when fired). If leaving on their own terms, I think there is still some level of responsibility that the PM has to take to ensure the completion of the project. Whether that be work side by side with the new PM, or prevent the change until after the project.