When I tried to include a RSS feeds web part to my public sharepoint site just the way I had done many times on intenal sharepoint sites, I found out that the RSS viewer web part was not available.
After searching the internet I found out why this was the case and I also found a workaround using a custom solution/webpart.
Even though it is possible to use RSS feeds this way on my public sharepoint site, it is not what I expected. In my opinion an RSS feed web part is such a fundamental part of websites nowadays that it should be available by default in Office 365 / Sharepoint online.
I hope that this will be corrected in the upcoming release of sharepoint (2013) for Office 365, but the preview still doesn’t provide this capability unfortunately:
blog.bjornhouben.com 
JV
April 1, 2013 at 22:14
I’m just starting to learn SP, and making a public site on O365. Stunned that there’s no simple RSS feed capability. Wow. Actually, I need a web part that will aggregate MULTIPLE rss feeds. So far not seeing anything that works on 2013.
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Bjorn Houben
April 1, 2013 at 22:18
If you find another way please let me know.
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sonia
February 16, 2014 at 00:20
Hi guys.. just tried something else. You go to your sharepoint site / site settings / under Web Designer gallery you will find web parts. Look for rss feed webpart and save it on your computer. Go to your public site and add a web part. Under the list of web part available you have the option to upload a web part,. Upload the rss webpart. Then setup the parameters and it works…
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Matthew DiLiberto
February 21, 2019 at 02:13
With SharePoint Online and the introduction of Modern Site pages, users now have the ability to add the Modern RSS Web Part directly to site pages.
This webpart currently has limited functionality compared to it’s predecessor, however, it is a good start for pulling a feed of external content onto a SharePoint page.
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Bjorn Houben
March 8, 2019 at 08:37
The problem back then was to add the RSS feed web part to the public website , but they killed the public site feature anyways 🙂
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