Reasons for long peer review times
As an editor, I can say that although we ask for reviews within a certain time-frame, we almost always get some reviews back late. But what often takes more time is finding willing reviewers in the first place. My journal aims to get four reviews for each manuscript, but usually settles for three. I start by sending out requests to five potential reviewers. Two weeks later, I might have accepts from two potential reviewers, declines from two, and no reply from number five. I'll send a reminder to potential reviewer number five. A week later, still not having heard back, I cancel that request and invite another two potential reviewers. Two weeks later, if I haven't heard back, or if they both decline, I'll look for more possibilities. Sometimes I get three reviewers on the first try, but sometimes, the process takes many weeks. We give our reviewers 21 days to return the review. In maths, I understand that reviewers are given longer, because they are expected to check the proofs in detail. Regardless of how long they are given, it is normal for some reviews to be late. Editors send out a reminder. Then a second reminder. Some reviewers, despite having agreed, never reply. In that case, we need to start the whole process again. This is why some manuscripts take so long to review.