Adobe Flash is proprietary software and hence Fedora does not include it by default.
Beware!
As of 2017, most websites don't need Adobe Flash any more (actually in 2018 less then 5% of websites uses it). Videos should play fine by default e.g. on youtube.com, vimeo.com, imdb.com or any social media. If they don't, try installing additional video codecs.
Please note that Flash and all its replacements are very broken and risky to use for stability, security and safety reasons. Here you can find statistics about vulnerabilities found and published for Adobe Flash: https://www.cvedetails.com/top-50-pro... - yep, it's the worst, it has the biggest amount of the most severe bugs and the highest weighted average of vulnerabilities and their severity (simple rating of how unsecured it turned out).
Chrome: Pepper Flash
Google provides newer features in Pepper Flash Player. Pepper Flash Player can currently be used with Chromium (and with Chrome); and Firefox with Freshplayer Plugin. Chrome dropped support for 32bits (March, 2016); therefore the support with Pepper Flash Player for 32bits.
Firefox, Epiphany, …: NPAPI Flash plugin
Note that this plugin is not supported by Adobe very well, it did not receive even critical security updates for a long time. You should not use it. This is one reason why Firefox will remove any support for Flash (and similar plugins) in the end of 2017.
Adobe Flash is known to be horribly insecure by design, so be very careful where to use it. Except for some very old or shady websites you probably don't need Flash at all. Especially on shady websites, please note that Adobe Flash is used as a primary target for malware, that is software attacking your computer and trying to take control over it.
Note that the download links below are insecure, since data transmission is not signed or encrypted. Make sure that the files are not manipulated. If you don't know how to do that, don't install them.
You need to set up Adobe repository first. This is used for automatic updates.
For the x86_64 (64-bit) version, click here or use a terminal:
sudo dnf install http://linuxdownload.adobe.com/adobe-release/adobe-release-x86_64-1.0-1.noarch.rpm
For the x86 (32-bit) version, click here
or use a terminal:
sudo dnf install http://linuxdownload.adobe.com/adobe-release/adobe-release-i386-1.0-1.noarch.rpm
You then need to install the plugin from the repository:
sudo dnf install flash-plugin
Checking the plugin in Firefox
Type the following text in the Firefox address bar:
about:plugins
Also check the Adobe Flash Player test page
Chromium web browser
Since Chromium 34 does not include support for NPAPI. Therefore, you need to use Pepper Flash to be able to use Flash.
Some extra steps are required to use the plugin with Chromium. After you've installed the plugin as detailed above:
On 64-bit:
1) wget -c -P /tmp/ https://dl.google.com/linux/direct/google-chrome-stable_current_x86_64.rpm
2) cd /tmp
3) rpm2cpio /tmp/google-chrome-stable_current_x86_64 ...
(more)
Better don't install it. NPAPI plugins are depreciated by most browser (vendors) as they are mostly obsolete and only pose an additional security risk. YouTube can be used without flash for years. It uses HTML5, available in every modern browser. (Not to mention that flash is obviously closed source.)