Today’s bumper covers are held on by a variety of hidden fasteners, self-locking tabs, screws and bolts. Depending on the manufacturer, you may see variations in style and location so you’ll have to do a little hunting to find all the fastening points.
There are typically a combination of round plastic fasteners, bolts and screws under the hood or trunk (depending on whether you’re working on a front or rear bumper), along the wheel wells and underneath the car. Once all this is removed you should be able to pull off the bumper cover with a few tugs. If it feels stuck, look for additional fasteners you may have overlooked or a push-tab that is still engaged.