How does a car behave when it has transmission issues?

If a car has an automatic transmission, potential problems include:

• not shifting into forward or reverse
• not shfting automatically into higher gears when vehicle speed increases
• not shifting automatically into lower gears when the vehicle slows down
• not downshifting to a lower gear to produce more power when a driver pushes down quickly on the gas pedal
• not shifting into neutral or "Park" positions
• not starting in the right gear, or skipping gears, or failing to shift to Top Gear