As the three finalists for Boston superintendent begin to make their case Monday to lead the nation’s oldest public school system, there are numerous big issues to watch out for.

The challenges facing the eventual winner are immense: deteriorating school buildings, declining enrollment, widely uneven school performance, persistent achievement gaps among students of different backgrounds, limited financial resources, graduates who struggle in college, students who don’t make it through high school, families who bail out of the system, schools at risk of state takeovers.

All of which raises a critical question:

What kind of superintendent does the 56,000-student system need to guide it through these thorny issues?