Version 3.7.1 fixes several minor issues that arose in the recent version 3.7 release, including some issues with the new auto-update feature. The official announcement of version 3.7.1 lists the changes.
The release of WordPress 3.7.1 provides a useful test of the new auto-update feature. I administer five WordPress sites, which I updated to version 3.7 the day it became available. Of those five sites, only two have updated themselves to 3.7.1 in the two days since its release. I will continue to update this post as the other three update themselves. Then I’ll decide whether to leave auto-updates enabled or continue to handle updates manually. Update 2013Nov01: two more sites updated themselves in the last day or so. One remains at version 3.7. Update 2013Nov04: one of the sites never updated itself, despite passing the auto-update tests. I updated it manually. I’ve concluded that the auto-update feature is useful, but not to be relied upon – at least not yet.
There have been a lot of reports of problems with the new auto-update feature. Most of these problems relate to hosting providers and limitations they impose on WordPress sites. Some of those problems were resolved in 3.7.1. In any case, you can diagnose auto-update problems using the new plugin Background Update Tester.
Another new plugin named Update Control allows you to control the way auto-updates work, including disabling them completely.
WordPress Tavern has a useful post about the new auto-update feature, titled “WordPress Automatic Updates – No Options For You!” There’s also a post on WordPress.org: “The definitive guide to disabling auto updates in WordPress 3.7.”