Many people took advantage of their warranty during the first round of Xbox 360 shipments.
Unfortunately the high demand was a bit overwhelming for Microsoft's initial release date and some hasty design decisions resulted in an unexpected operational error.
This has been resolved and is no longer a problem with new models, but here's how to fix your Xbox if you have encountered the infamous Red Ring of Death, and no longer have a warranty.
Overheating of the console, for several reasons, was the main malfunction behind the unfortunate error.
This overheating was a result of eroding soldering joints and screws that were heat-sensitive.
Obviously, the way to fix this is to replace these parts with stronger, less-sensitive ones.
Remove the shell of your Xbox and locate and remove the motherboard.
Flip it over and you'll see two X-clamps which house the heat sink.
Carefully remove these three parts and clean them thoroughly.
Occasionally, the manufacturing process of the early models would leave vestiges of foil on the heat sink that would interrupt its electrical conductivity.
This, of course, resulted in overheating.
Once the heat sink is clean, apply thermal compound to ensure it maintains insulation.
Under the heat sink, you will find the CPU and the GPU which you should also clean, even if with a few puffs of air.
When it comes time to reattach the clamps, be sure to replace the screws with stronger ones, as the original screws would erode from heat exposure, weakening the overall integrity of the console construction.
With all the necessary parts cleaned or replaced, this should do the trick.
If you still have problems, you may need to replace the GPU itself, as early models used an in-house design that unfortunately also overheated too quickly.
This will be the most costly step of your repair, but hopefully it's a step you won't have to take.