I’ve been using SMS Backup + (by Jan Berkel) for over 5 years now.
It’s one of the very first apps I install whenever I get a new phone or flash a new ROM.
It allows your to backup your SMS, MMS and call logs to your Gmail account.
It’s completely free of charge, doesn’t have any shady permissions and is AD free.
On the Play Store it lists that it has “in-app purchases” that is ONLY if you want to donate to the developer.
WORST REVIEW EVER!