Roomba Running Forward in Circles

Usually is a wheel problem. Often it's just the wheel sensor.
Check diagnostic modes 5 and 6. Do both wheels move.

Yes both wheels move:
check mode 6 for flashing lights

Yes, both wheels move, both lights flash:
go back to mode 5 and check how straight of a line it moves. If the circling is caused by one wheel turning faster than the other, you may have a slipping belt that needs to be replaced. (See: Roomba Belt Replacement.).

No, both wheels move, but one or both lights are shining steady:
Optical sensor on the sides non-flashing are blocked with dust and fuzz. You may be able to clean them out with compressed air, or you can partially disassemble and pick it out with a tweasers. See Roomba Belt Replacement for how to open it up

No, one or both wheels aren't moving:
Can you hear the motor turning despite the wheel not turning?

Yes, the wheel isn't turning but I can hear the motor turning.
You probably have a broken belt. See Roomba Belt Replacement.

No, the wheel isn't turning and neither is the motor.
You have a dead motor. You need to talk to Roomba customer service.

Roomba Running Backwards in Circles

Usually the front bumper or a cliff edge detector problem. Again, usually solved by removing the stray fuzz.
Check for proper operation.

Roomba Ignoring Virtual Wall Unit

Check if the Virtual Wall Unit is detected by the Roomba in diagnostic mode 5.

Roomba Not Cleaning Edges of Walls

Usually this is NOT the wall brush, but a bad adjustment of the wall following mode. Use the wall following adjustment screw to set to the proper distance.

See also the articles: Roomba Diagnostic Songs, Roomba Discovery Diagnostics, Roomba Belt Replacement.