Door closers
The closer controls how fast the door swings and how hard it latches. A leaking or badly adjusted closer is the most common reason a door slams or stays open. Adjusting or replacing it fixes both security and the slam.
Opening force and accessibility
Interior doors on an accessible route should open with no more than about 5 pounds of force under the ADA standards. An over-tightened closer is a common reason doors fail this. See the US Access Board's ADA standards for the details.
Hinges, pivots and sagging doors
Heavy doors wear their hinges or pivots. A sagging door drags, rubs and stops latching. New hinges or a pivot and a re-hang put it back in line.
Locks and levers
Commercial levers are graded by durability. Grade 1 or 2 hardware stands up to heavy traffic. Residential-grade locks on a busy door wear out fast.