For publishers considering FeedBurner, follow these two Google tutorials for redirecting your Blogger feed to FeedBurner:
Redirect Blogger Feed to FeedBurner (redirect instructions are after the FeedBurner set up information).
Should I Change My Chicklets to FeedBurner, too? (instructions for using your original feed address in the event FeedBurner shuts down - shows how to keep your subscribers)
Using Google's Feedburner for Feeds
Some viewers use RSS readers to aggregate the articles they receive from various websites. RSS feeds can be read with email, torrent, or aggregator services which receive content from many websites for delivery to one location. In some ways social media has helped to make RSS feeds "obsolete" yet, as Frederic Lardinois (@fredericl) pointed out, RSS is still used regularly on the backend of sites by publishers. FeedBurner was once a leading management system for RSS feeds, as Lardinois has also pointed out, over the last few years Google has ended services associated with FeedBurner (AdSense for Feeds, etc.).
The two articles written by Lardinois were written in 2012 and 2013, it is currently 2015 and FeedBurner is still a usable product and the feedburner forum shows that people still seek and receive comunity assitance in utilizing the tool. Lardinois also reminds us that Google didn't update the service when the rest of their sites got a face lift. So, speculation is that the service is nearing it's end of life. Which is possible, entirely possible.
In brief: people still use FeedBurner for RSS feeds; Google has allowed the feed service to remain while depreciating the API; and, publishers must decide whether to remain on FeedBurner or use an alternative.
Personally, I enjoy using FeedBurner and I redirected my Blogger sites to my FeedBurner account a while back. I will continue to utilize FeedBurner as long as Google continues to host the RSS management service. Should Google decide to end FeedBurner, I'll update this post and create instructions for switching over to another RSS management service.
NOTE: Whatever you do, don't delete your FeedBurner feeds. Deleting your FeedBurner feeds will allow your feed to be taken over by another user...which means your users could receive notice that you're no longer posting at the feed's URL, then the subscriber may begin receiving feed content from whomever took over your former feed URL - for more information read this WPBeginner article.
Here are a couple FeedBurner alternatives that I have not tried out. If you try them out, please leave a comment with what you like/don't like about them. :D I may try them, later.
For kicks: here's an RSS Seach Engine