How do I restore (downgrade/upgrade

How do I restore (downgrade/upgrade) my device to a non-latest iOS version above iOS 5.0?
Information for iOS 6.x, 7.x, or 8.x on iPhone 4S, iPhone 5, iPhone 5C, iPhone 5S, iPhone 6, iPhone 6 Plus, iPad 2, iPad 3rd generation, iPad 4th generation, iPad Mini, iPad Mini 2, iPad Air, or iPod touch 5th generation
If you have one of these devices on iOS 6.x, 7.x, or 8.x, there is currently no way to downgrade, upgrade, or restore it to any version of iOS that isn't the newest available version of iOS, even if you have SHSH blobs. You can only restore it to the newest available version of iOS.

(The exception is the iPad 2 - if you have iOS 4.x and 5.x SHSH blobs saved, you can downgrade it from iOS 6.x to 5.x using redsn0w. Keep reading for instructions on this.)

Information for iOS 7.x on iPhone 4
If you have an iPhone 4 on iOS 7.x, there are no tools available for downgrading, upgrading, or restoring it to any version of iOS 7.x that isn't the newest available version of iOS, even if you have SHSH blobs.

General information on requirements for downgrading and upgrading
What do I need to find out first? To successfully restore to a specific version of iOS, the first step is that you must have that specific version's SHSH blobs already saved for your device, either on Cydia's servers or on your computer (although this is not a guarantee that you'll be able to restore to that version). Here's a guide to understanding what SHSH blobs are and why they are required. In short, if your device was previously jailbroken (or has been jailbroken for a while), Cydia may have been able to save SHSH blobs for some iOS versions for your device. You can check a device's SHSH blobs saved on Cydia's servers by running TinyUmbrella, plugging your device into your computer, clicking the "Advanced" tab, checking "Request SHSH from Cydia", unchecking "Set Host File to Cydia on Exit", and then clicking "Save SHSH" at top right. If you have the SHSH blobs you need, follow the instructions below for your device type.

Can't I just change my hosts file or run TinyUmbrella to restore? Nope. The TinyUmbrella TSS Server procedure does not work anymore when restoring to any version of iOS 5.x or later, because Apple has introduced an APTicket/APNonce check for restores to iOS 5.x and later. Instead, you need to use redsn0w to create and restore to a custom IPSW (custom firmware file) with stitched SHSH blobs. There is no need to change your hosts file anymore; that'll just cause problems.

Instructions for iPhone 4, iPhone 3GS, and iPod touch 4th generation restoring to 6.x
NOTE: If you have iTunes 11.1 (or later) installed and you are restoring an iPhone 4, you need to first downgrade iTunes to version 11.0.0.
Uninstall Apple Application Support, Apple Mobile Support, iCloud and iTunes 11.1 from the Windows control panel, or delete iTunes from the Applications folder on a Mac.
Rename iTunes Library.itl to iTunes Library.bak. This file is located in C:Users[your username]MusiciTunes on Windows. On a Mac, it's in ~/Music/iTunes.