Mozilla Firefox 44 now supports Push Notifications
The Mozilla Blog;
Firefox for Windows, Mac and Linux now lets you choose to receive push notifications from websites if you give them permission. This is similar to Web notifications, except now you can receive notifications for websites even when they’re not loaded in a tab. This is super useful for websites like email, weather, social networks and shopping, which you might check frequently for updates.
This is great for the web/desktop version of our favorite websites, not sure though for mobile, since most (major) website already have their on app or users are either just using Safari for iOS or Chrome ...