How do these work? I see it isn't to do with level, or at least not just that. So how do you raise the 'star' rating of a monster you already have?

Perhaps you should visit the WIKI, it has a lot of relevant information for new and experienced users. You should also use the search feature here on the forums to search instead of creating all these new topics. Many of these questions have been asked and answered.

@mistr-bill said in Monster library star ratings:

How do these work? I see it isn't to do with level, or at least not just that. So how do you raise the 'star' rating of a monster you already have

Hi . . . I guess you're referring to "ranks". . . You need to rank up your monsters according to which chapter in the book you want to get, for example: if the chapter requires monsters with 1then you have to rank up those monsters to level 105 and so on in order to get the rewards mentioned in that chapter. You can rank up monsters with cells in the Lab, whether is through your teammates' cells donations/requests, with Elemental cells (in case of Legendary or Elite monsters) or with Elementium. Bear in mind that for Permanent Books, you get rewards for ranking but for Time-Limited Books, ranking up doesn't give you any rewards