It is doable, but it won't be exactly like it was. And really, you don't want that. You'd like to have a better relationship. The old relationship included affairs and other dysfunction.... you want a new and improved version of the old you. It is doable, but it takes a lot of work.... a lot of time.... a lot of commitment.... a lot of crying.....
The commitment on both of your behalves needs to be ever present. A level of vigilance and an ever presence of self and accountability will be necessary. All other people aside, this is about the two of you. Anyone can say whatever they want about a situation... but this is your situation, making it different from every other situation out there. For some, infidelity is a deal breaker. (Both my wife and I thought that, until I committed adultery and we decided to work on it.)
It won't be easy, but nothing worth fighting for is easy. Seek a therapist, do a ton of research. You will probably find some very incredible information on this board or elsewhere within
medhelp.org. (I know I did, and still do) Ask your BF to look into medhelp and create his own profile and develop his own dialogue. (With him being the cheater, he probably won't be too well received in the beginning, but some will come around.... you'll/he'll be amazed by all of the similarities with all of the stories on the board and how they mirror yours>)
Good luck, stay strong and united and both of you know you are doing what is necessary to make it work.