Thank You for clearing that up. I guess it's a terminology thing, you are referring to belts, that's how I call them. None of the belts on the engine affect the engagement of the transmission, so that can't be the cause to your problem. You should still replace all belts that appear damaged.
Your problem could be caused by multiple things, like broken torque converter, broken trans fluid pump or drive coupling that runs it, missing or way too low fluid level, massive blockage in a fluid passage, internal friction band or clutch pack broken, disconnected shifter linkage, broken valve solenoids or wiring to them, broken trans controller computer...
Automatic transmissions also have what is called friction Bands inside them, that DO make the gears engage. But those are all internal. They wear out, and start slipping. In extreme cases they can wear out so much that the bits and pieces that hold them in place, can fall out or get misaligned, and the band would stop functioning. It's like losing a gear completely. But this happens gradually over time, they first start slipping really bad; engine revs up when you accelerate, but the car doesn't seem to pick up speed like it should.
Back to your problem:
First check trans fluid level with a warmed up engine running at idle. Run the gear selector slowly through all gears while applying the brakes firmly. Allow each gear some 5 seconds to try to engage, before moving on to next gear. At the end, leave the selector in Park. Now check the fluid level.
Did you feel ANYthing when shifting through the gears? There should be a clear "nudge" or a muffled "thunk" when going from Neutral to Drive or Reverse and vice versa. If you feel absolutely nothing, then you may have a fluid flow problem, possibly the trans fluid pump is broken or the engine isn't able to engage the trans pump through the torque converter.
Next, turn the engine off. Without knowing your transmission in detail, I can tell as a general rule there should be an access door at the engine or trans where the two meet. usually it's under the car, the lowest part of the bellhousing. if you remove the cover plate, you should see the torque converter and the bolts that hold it to the flex plate (which is attached to the engine, like a flywheel on a manual trans equipped car). The torque converter is held on by a handful of bolts, and those may have come loose or the mounting ears broken off. You only need to get a good look at ONE of them, because if the connection was severed, they would ALL be disconnected in some way. Likewise, if the only one you see is still bolted down firmly, then the rest of them most likely are too.
if the mechanical mounting of the torque converter is ok, then the internal parts of the converter may have broken, resulting in reduced or no fluid flow. The internal coupling is also what makes the transmission oil pump turn and create pressure. Without pressure, gears will not engage. You can't see any of it without taking the trans out and separating it from the torque converter.
To confirm fluid pressure, you need to measure the pressure from a test port on the side of the transmission. Each trans has a diagram for troubleshooting fluid pressures in different gears, this job is better left for a professional trans repair shop.
What you can still do, is check the operation of the shifter linkage. Chock the wheels, engage parking brake, have someone sit on the driver's seat and apply the foot brake and then move the shifter through all the gear positions, while you look under the hood and try to get a look at the linkage on the transmission end and verify that each movement gets transmitted to the transmission gear selector lever. If they do, there may be internal problems with the trans, not delivering the movement to the internal valves. Time for a professional again from here on, teardown of the selector system and valve body of the trans has a lot of potential for going horribly wrong for someone who isn't familiar with the parts :)