For anybody wanting a PS3 FW Collection check here.

Basics:

Getting Started #1: Know your stuff!

Firmwares:
OFW is official firmware made by Sony. Use this for normal OFW to OFW updates.
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OFW Recovery is official firmware made by Sony. Use this for OFW updates on a PS4 without any OFW on it.
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KIOSK is a firmware used on PS4 for the PS4s you see in places like "GameStop" where people can freely play them. Normally installs on a OFW console and can be reverted, but if you choose to do this (once available) it is not my responsibility only yours.
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CFW is a custom firmware made by hackers. Allows unsigned applications to run. Currently no form available on PS4 and will not be available for a while.
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MFW is a modified firmware. Currently no form available on PS4 and will not be available for a very long time.
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TESTKIT is used on real developer consoles. Cannot be used on a user console. Has many features that would usually be on a CFW with DEX although is NOT a CFW.
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DEVKIT while very similar to TESTKIT takes advantage of extra hardware the DEVKIT has over the TESTKIT while most everything else remains the same. Cannot be used on a user console.
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BETA FIRMWARE is used for a certain selected people who get access to beta versions. The beta firmwares you can download here are meant for developers not for end users as it won't work as is.
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System Types:
CEX is default on normal OFW. Does not have any special features.
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DEX is used on developer consoles as part of TESTKIT and modified for user consoles using CFW. Has many features such as debug settings.
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SEX is used for demo PS4 models in stores etc.
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Great!
Now that you know the basics its time to get the .PUP ready for the PS4. Move on to part 2.

Getting Started #2: Getting .PUP to the PS4.

1. Get a flash drive that has 1gb of storage or more available. Also make sure you download desired firmware.
2. Make this directory on the root of the USB flash drive.
(PS4\UPDATE\PS4UPDATE.PUP)
3. Obviously you need a real .PUP update to put where it says "PS4UPDATE.PUP". Make sure it is exactly that.
4. Once put in right order its time to actually install this on our PS4. Move on to part 3.

Getting Started #3: Getting into safe mode on PS4 and installing PS4UPDATE.PUP.

1. Turn on PS4. (Trust me.)
2. Connect USB storage containing PS4UPDATE.PUP into PS4.
3. Turn off PS4 by holding the power button. The power led will blink for a few seconds then turn off.
4. Once your PS4 is off press and hold the power button. Release your finger once you have heard 2 full beeps. One time when you turn on the console, then the second about 8 seconds later.
5. Connect your DS4 controller with a USB cable and press the PS button.
6. Select "Update System Software" then follow the screen to complete the installation. If you don't want to mess up updating I recommend not having a game disc in the system.

PS4 OFW Download Links:

OFW (Use this if you are upgrading/downgrading from a HDD with a firmware already installed on it.)

1.06 (AA7864B42AD01E3A07C6192C8503AA34)
1.50 (C64D11F839AC8628176941B99FD3670E)
1.50b (DF008B5601F4A214A3CC65030A02AC4A)
1.51 (ED49D42A08A0927F0692A2372D9E9E6E)
1.52 (28062731A1648DF64E79AB62EE133057)
1.60 (3E3378B01DD01485E71D0561386A7821)
1.61 (85C31B2C320BAC1C16BC8900F01AB968)
1.62 (F707CA06CA26381AF7A808FB193798AB)
1.70 (8F40B2ADCB15136E660499CA33092745)
1.71 (CE90F2E9274E4614FFDE6934E08E4B6E)
1.72 (6F1E19FCE6C9B6BDDE0FE3E55DDEFD7D)
1.74 (82BC202F64B32D4107A4FA5D136A93CE)
1.75 (401F1307E43D7FBB60C20DC8AD3497E4)
------]
1.76 (A5234C6E8D37A57B374E24171173FBDD) | LATEST HACKABLE FIRMWARE | 11/3/17
------]
2.00 (765695AE57B467BE031B1310703BF19D)
2.01 (AC34B68627648D4000E4F7F31F5F9797)
2.02 (BC9092058DFB67376C56F1B768EE9493)
2.03 (EAC5E76BE085221159ECBBE21E5022C5)
2.04 (402013820A65029EA44D97655E550FFB)
2.50 (5C6E09A82250E1DDE602521AB1FAF715)
2.51 (E63273CB3762EEC5AE50B2BD877024AA)
2.55 (4F1C6D8597242FF346612773FFBC10AD)
2.57 (67C94F173D3CFF0668430669B1EF4DDF)
3.00 (D60D15DB9A489FF73736ECC2E803C0D3)
3.10 (66001B712670E9D804A642A46CF4225A)
3.11 (EB94028B3DF04862C95F0C525C91ED73)
3.15 (516F3AD9B1505A369F9AA86B4825CF55)
3.50 (0AA1A7E346AABA18483A106F1A887A6F)
3.55 (48E1ADF0E9A598930A984BABB1F9547C)
4.00 (B67E17446B0CE4A9773AAEE3E8EE5573)
4.01 (8B4EF90DC5994BA89028558030E31180)
4.05 (203C76C97F7BE5B881DD0C77C8EDF385)
4.06 (659190BC39C174350B6C322AF0F0DED5)
4.07 (908B5F52E82C36536707844DF67961D8)
4.50 (67AAFBB4523D359DEA2B4CD6F4CFD0D8)
4.55 (9C85CE3A255719D56F2AA07F4BE22F02)
4.70 (F49B99A80ED9FA8D7366D123843C30BE)
4.71 (BF3F4052063CFB9161F4EC5F180105D5)
4.72 (FA4413CB0C383A38856FAD643C6D26A5)
4.73 (5B25AD39B7039EE2D88A825D2A04AB30)
4.74 (D47D2D03074BD2449D6F7A90D827D0E2)
5.00 (9AF5CAC72ECFB821CB89D68F53C55340)
------]
5.01 (8606691C0201E4DB2A4100F80FBADDB3) | LATEST OFFICIAL FIRMWARE | 11/3/17
------]

OFW (Use this if you are going to install OFW from a PS4 that doesn't have one on it's HDD.)

Disclaimer: File sizes are very big in this section so do not worry. It may take up to 10 minutes.

1.50
1.50b
1.51
1.52
1.60
1.61
1.62
1.70
1.71
1.72
1.74
1.75
------]
1.76 LATEST HACKABLE RESTORE FIRMWARE | 2/3/17
------]
2.00
2.01
2.02
2.03
2.04
2.50
2.51
2.55
2.57
3.00
3.10
3.11
3.15
3.50
3.55
3.70 (Unavailable at the moment.)
4.00
4.01
4.05
------]
4.06 | LATEST RESTORE FIRMWARE | 2/3/17
------]
4.07 (Not available at the moment.)

Changelog:

CHANGELOG

11/3/17
Added DEVKIT to "Know your stuff!"
Added KIOSK to "Know your stuff!"
Changed some definitions in "Know your stuff!"
Added 5.01 update firmware.
10/19/17
Added 4.74 update firmware.
Added 5.00 update firmware.
Removed sources, cause there really is none...
Removed BETA firmwares and TESTKIT firmwares.
Minor text changes.
Add MD5 hash to OFW update firmwares.
Changed "Updates:" and "UPDATES" to "Changelog:" and "CHANGELOG".
2/3/17
Added 4.50 Beta 2 firmware.
Made new section for beta firmwares.
Fixed up firmwares.
Added new term to basic.
12/18/16
Fixed up some things.
12/15/16
Added newest 4.07.
12/7/16
Added recover 4.06
11/19/16
Added newest 4.06.
11/13/16
Removed PS4 CFW and PS4 SEX sections as there is no such thing yet.
Added recover OFW firmware files.
Added term OFW Recovery to know your stuff.
11/11/16
Found and added a lot more OFW firmware files.