If your freezer is not freezing or staying cold, here are a few of the most common causes and their resolutions. Video
 

The condenser coils (typically found on the bottom of the refrigerator) can gather dust and grime over time, which can prevent the coils from dissipating the heat and allowing the unit to stay cool. These coils will need to be cleaned if accessible.

Similarly, if the condenser fan motor is malfunctioning, the motor will not draw in air to cool the condenser coils and the compressor. Confirm there is not an obstruction preventing the fan blades from moving. If not, the motor bearings are likely worn out, or the motor itself is not allowing a continuous electrical path. This can be tested with a multimeter. If the motor itself is faulty, this will need to be replaced.

The evaporator fan motor draws air over the evaporator coils to remove the heat from the air. A faulty evaporator fan motor may result in the freezer not freezing. As with the condenser fan motor, check for an obstruction of the fan blades, the condition of the motor bearings, or the electrical path to the fan motor. Replace the motor if needed.

The defrost heater may have an electrical issue preventing it from turning on. This would cause ice buildup and a freeze-over. Test the wiring with a multimeter to see if there is a continuous electrical path present in the part. If the heater is found to be faulty, it will need to be replaced.

The thermostat control in the freezer may register an incorrect temperature, turning off the motor(s) before it is needed. Use the multimeter to test the electrical current. Double check the manufacturer's recommended temperature - the multimeter will not register a continuous electrical path once the freezer is at this temperature.

If the refrigerator has a user display/control board on the front, this could send incorrect signals to the refrigerator, preventing the freezer from functioning correctly. This can be replaced if necessary, but proceed with caution.

With all replacement parts, please remember to check the age of the unit and compare the price of the repair vs. total replacement.