Edit by Moderator on November 2017:

PS3 Custom firmware is still possible using the old method which requires an official Firmware v3.55, with some limitations. See the spoiler from November 2012 below.
PS3 Custom firmware is now also possible with the official firmware v4.82, but currently works only on specific consoles.

Compatible PS3 consoles:
PS3 Fat : ALL COMPATIBLE !
PS3 Slim : Compatible only if your PS3 console with serial number 2xxxx has the minimum installable firmware 3.56 or lower. If your PS3 console with serial number 2xxxx has minimum installable firmware 3.60 or higher it's currently not compatible.

NOT compatible yet:
PS3 Slim with serial 3xxxx
PS3 super slim (all of them)

The new hacking steps are:
1) First, you need to install the official firmware 4.82 (and only 4.82 !).
2) Then you can use a software method to edit your NAND or NOR chipset from a browser hack.
3) Once it's patched, you can install a CFW 4.82 (not lower).
4) Once you are on CFW 4.82, you are free to downgrade to another CFW version or keep the one you installed.

The main release thread and the full tutorial is available here:
https://gbatemp.net/threads/490200/



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Edit version by Moderator from November 2012:
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Status on November 2012

Edit by Moderator on November 2012:

PS3 Custom Firmware are now possible past fw3.55, but most (if not all?) consoles require that you are in 3.55 before installing a custom firmware.
Not all consoles allow downgrade, so before upgrading/updating your console, please read and understand if what you are doing is correct. you may be unable to downgrade without buying an hardware to flash your console's NAND/NOR chipset. I'm sure most of you don't know how to do that, so better safe than sorry!

Our current guide on GBAtemp covers old cFW installation (3.40 to 3.55).
There are some user's guide in the Guide and FAQ section, but I encourage yourself to read other website too and understand very well the options you have (there are a lot of cFW past 4.x, you need to choose one correctly).
You are also encouraging to post a guide to help your fellow Tempers here. The community should help each other.

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Original post by Nujui from 2010:
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Original post by Nujui from 2010

Right now, the only OFW able to have CFW is 3.55 and below. Anything above will not able to use CFW. The only way you can have CFW at this point in time is if you buy a downgrade, such as Progskeet or the E3 Flasher, to downgrade to 3.55.

When is there ever going to be a cfw pass it? Answer is: We don't know. There could be someone secretly working on a CFW as I'm typing this, if you really want CFW for 3.56 and above, then be patient.