[Branch Delay]

This software-only jailbreak only works for Kindle firmware version 5.6.5.
If you have a later firmware version, this jailbreak won't work.
If you have firmware version 5.6.5 or earlier, switch to aeroplane mode immediately, to make sure you don't get auto-updated before installing the jailbreak.

Instructions
If your Kindle firmware is earlier than 5.6.5, manually update to firmware version 5.6.5. See either this post or the spoiler tag at the bottom of this very post for links to the firmware update for the different models of Kindle.

Download the jb.zip file attached to this post. Unzip it, and you will have a folder 'jb' containing four files, 'jb', 'jb.html', 'index.html', 'frame.html'

Connect your Kindle to your computer via USB and copy the four files to the top level of your Kindle. Eject your Kindle.

Open the 'experimental browser' on your kindle and enter the address

file:///mnt/us/index.html

and then follow the instructions on the screen. When your Kindle complains about being in aeroplane mode, just cancel the warning, DO NOT turn on wifi.

Once you get the "Jailbreak succeeded!" message in the title bar, the first stage has worked. It's now necessary to add the code that will preserve the jailbreak in most circumstances. This is described in this post, but the instructions are also included here for convenience.

Download the JailBreak-1.15.N-FW-5.x-hotfix.zip file attached to this post, and unpack it.
Connect your Kindle to your computer again and copy the 'Update_jailbreak_hotfix_1.15.N_install.bin' file to the root directory of your Kindle.

Now, eject & unplug your Kindle, and go to [HOME] -> [MENU] > Settings -> [MENU] > Update Your Kindle.

Your Kindle should do a software update and then restart.

Now the jailbreak and hotfix are safely installed, you can delete the five files, 'jb', 'jb.html', 'index.html', 'frame.html' and 'Update_jailbreak_bridge_1.15.N_install.bin' from the root of your Kindle.

You will now probably want to install Kindle Unified Application Launcher (KUAL) and the MobileRead Package Installer (MRPI), since being able to install third-party software is the entire point of the jailbreak.
You'll want to use up-to-date versions of those from the Snapshots thread.

And then you'll be able to install any of the cool additions available in the developers' forum.

(Once the hotfix is also installed, you can turn off Aeroplane mode, and allow your Kindle to update to the latest firmware without worrying that the jailbreak will be erased. You'll need to reinstall any of the 'cool additions', but the jailbreak and hotfix will remain installed and won't need reinstalling.)

These instructions are derived from the simple install instructions in this post and this post.

Old instructions:

Spoiler:

[added by pdurrant]
First set of instructions and download at this post. Quite complex, so early adopters only, IMO.

Step by step instructions for simpler method now at this post by mj52.





FAQ by eschwartz: Read this at least once
Q: How do I prepare for this?
A:

1. Put your device in Airplane mode.
2. Update your Kindle to fw5.6.5
   • For the PW2: Update instructions Direct firmware link
   • For the PW3: Update instructions Direct firmware link
   • For the KT2: Update instructions Direct firmware link
   • For the KV: Update instructions Direct firmware link
3. Sit back and wait for a simplified jailbreak to be assembled, or get the current version HERE.

Q: Will this work on any other firmware version?
A: No.

Q: What do I do if my Kindle already came with a newer version of the firmware?
A: Newly purchased devices will not ship with the latest firmware. So even when an updated firmware is released, you should still be able to jailbreak.

Q: Will this work on the __insert_device_name_here__?
A: If it runs fw5.6.5. It has been tested and works on the PW2, PW3, KT2, and KV, so that's all of them.
English, Chinese, French, Italian and German have been tested, in case the language matters.