iOS 10.3.1
Apple suddenly released iOS 10.3.1 full version without any beta test. Most probably Apple target to kill Luca Todesco's Yalu103 iOS 10.3 jailbreak test from iOS 10.3.1 release. (Yalu103 iOS 10.3 jailbreak)
Yalu103

Yalu103 jailbreak is the upcoming semi-tethered method for iOS 10.3. Still, Yalu103 is beta testing and seems to be Yalu103 tool would release for public very soon. Read more about Yalu103 jailbreak.

Jailbreak without PC
There is two type of jailbreaking tools, * Computer support tool, and Online tool. Online jailbreak also calls as Without PC jailbreak. JailbreakMe / zJailbreak and Semi jailbreak are the most popular without PC jailbreak methods at this time. Read more about Without PC jailbreak.
zJailbreak

zJailbreak is the third party app installer method for iOS devices. At this time zJailbreak support iOS 10.3.1 to iOS 10.2 version. You can install unlimited apps and tweaks after install zJailbreak.

Install Cydia to iOS 10.3 - 10.3.2

Currently, iOS 10.3.3 is the latest iOS version, zJailbreak is the only method to install the Cydia for iOS 10.3 - 10.3.2. Unfortunately, You can download Cydia with limited features for these iOS version.
Step 1) Visit iOS 10.3.2 jailbreak page from your iOS Safari browser
Step 2) Install the zJailbreak  (you must enter your passcode to install it)
Step 3 ) Open the zJailbreak. Tap Cydia interested in installing it

Install Cydia to iOS 10 - 10.2 (All other iOS 10 versions)

You can use zJailbreak to install full functional Cydia for iOS 10.2.
Step 1) Install the zJailbrea just following above method (Cydia download for iOS 10 - 10.2)
Step 2) Install Yalu Jailbreak using zJailbreak app
Step 3) Open Yalu Jailbreak and tap "go" button
Step 4) OK, Your device will restart, and you can find the full functional Cydia on homescreen of your device.

Cydia iOS 10.1.1 , iOS 10.1 , iOS 10.0.1 , iOS 10

Yalu online tool and PC supports tool supports iOS 10 - 10.1.1 Cydia download. Visit iOS 10.1.1 - iOS 10 Cydia page for more details

Cydia iOS 9.3.5 - iOS 9.3.4

Full function Cydia does not support iOS 9.3.5 - 9.3.4 version. We recommended to upgrade your devices to the latest or use semiJB Cydia alternative. Go to iOS 9.3.5 - 9.3.4 page for more details.

Cydia iOS 9.3.3, iOS 9.3.2, iOS 9.3.1, iOS 9.3, iOS 9.2.1 and iOS 9.2

Pangu released tool for iOS 9.2 – iOS 9.3.3. But It only works with iPhone 6S, iPhone 6S Plus iPhone 6, iPhone 6 Plus, iPhone 5S, iPhone SE, iPad Air 2, iPad Air 1, iPad Pro, iPad Mini 4, 3, 2 and iPod touch 6G. Rest of all devices can use SemiJB Cydia alternative. Visit iOS 9.3.3 - iOS 9.2 pages for more details.

Cydia iOS 9.1 – 9

Pangu9 untethered jailbreak tool supports iOS 9.1 - iOS 9 Cydia download. Semi jailbreak partial Cydia also available for iOS 9.1 - iOS 9. Go to iOS 9.1 – 9 page to get most compatible Cydia install method.

Cydia iOS 8.4.1

No untethered or semi-tethered jailbreak method for iOS 8.4.1 version. Semi jailbreak partial Cydia is the only available method for iOS 8.4.1 version. Go to iOS 8.4.1 page for more details.

Cydia iOS 8.4 – 8.1.3

iOS 8.1.3 – 8.4, you can use the following tool to download and install Cydia to your iDevice. Make sure that you download appropriate tool for your operating system of your PC. Windows users can use TaiG tool, and Mac users can use TaiG/PPJailbreak. Tap above detect button from your iOS device or visit iOS 8.4 and iOS 8.3 pages for more details.

Cydia iOS 8.1.1 – 8.1.2

iOS 8.1.2 - 8.1.1 users can use TaiG jailbreak tool and PP jailbreak tool for download Cydia. TaiG only supports Windows platform and PP only for Mac. Visit this page from your iOS device to get most compatible Cydia install method or visit iOS 8.1.1 and iOS 8.1.2 page for more details.

Cydia for iOS 8 – iOS 8.1

Pangu jailbreak tool supports iOS 8 - 8.1 to download Cydia. Both Windows and Mac users can use the tool. Visit iOS 8.1 - iOS 8 page for download Pangu jailbreak tool

Cydia  iOS 7.0 – 7.1.2

Pangu jailbreak tool supports iOS 8 - 8.1 to download Cydia. Both Windows and Mac users can use the tool. Visit iOS 8.1 - iOS 8 page for download Pangu jailbreak tool

Cydia iOS 6 – 6.1.6 (For iPod touch 4 / iPhone 3GS)

Both Windows and Mac users can use p0sixspwn v1.5.3 to install the Cydia for iOS 6.1.3 – iOS 6.1.6 (including 6.1.3, 6.1.4, 6.1.5, 6.1.6). These tools install Cydia automatically after the jailbreak process.

Cydia Impactor is a GUI tool and Windows / Mac / Linux users can install IPA file by using Cydia Impactor tool. Formerly Cydia Impactor was also known as Cydia eraser. Cydia eraser helps to unjailbreak your device without loosing jailbreak ability.
When you are using Cydia Impactor, you should use Cydia Impactor latest version. So you can get most recent Cydia Impactor version by taping below links. Otherwise, follow Click Check for an update as shown below image.
Download Cydia Impactor
Mac OS X, Windows, Linux (32-bit), or Linux (64-bit).
Cydia is the most popular 3rd party application manager for iPhones, iPads or iPod touches. Saurik is the developer of this app manager. Here is the official website of the Cydia.
Cydia downloading methods vary according to device models and iOS versions

Cydia to iOS 10.2.1 - iOS 11

Users must Jailbreak their devices to download Cydia to their devices. However, Jailbreak is not available for iOS 10.2.1 - iOS 11. Therefore some Jailbreak apps may not work with iOS 10.2.1
Watch the following video to find Cydia installation method and Cydia features for iOS 10.2.1 - iOS 11.
Refer iOS 11 Jailbreak page for more information.
This video shows the Cydia installation method for iOS 11.

You can download the Cydia to iOS 10.3.3, 10.3.2, 10.3.1, 10.3, 10.2.from this method
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