We've established that a map and compass are necessary for navigation, providing vital information about where you are, where you want to be, and what is between you and that goal. Unfortunately, sometimes these vital tools fail: maps can become wet or torn or otherwise become unusable; compasses can break or be lost. Losing your tools makes things far more difficult, but it isn't the end of the world.
If you find yourself without a map in hilly or mountainous territory, "downhill and downriver" is a good rule of thumb -- work downhill until you find the water, then follow the water downstream to the people. In flatter, more arid territory, look for stands of trees or bushes that mark watering holes and streams, and follow the water from there.
If you lose your compass, there are ways to find approximate directions without it. A rough East-West bearing can be found early and late in the day simply by checking the sun. The closer you are to the equator, the more accurate this bearing will be, but even at my 41 degrees north latitude, it's close enough to work.
Another method involves standing a stick at least 12" long vertically into the ground. Mark the tip of the shadow cast by this stick, wait about 15 minutes, and then mark the new point of the shadow tip. A line drawn between both marks will be a quite accurate East/West indicator. The first mark is the western end of the line, the second mark shows east.
At night, the stars are a time-tested method of navigation. In the Northern Hemisphere, Polaris (the "north star") is the star of choice. Learn to locate it, and you can draw bearings without the sun at all.
Tools are a wonderful thing. Being without them is tough, but with this knowledge and some practice, you'll have no excuse for being lost.