If you ask a room of developers what mobile device features are missing from the web, push notifications are always high on the list.
Push notifications allow your users to opt-in to timely updates from sites they love and allow you to effectively re-engage them with customized, engaging content.
As of Chrome version 42, the Push API and Notification API are available to developers. And other browsers will follow.
Sending notifications to mobile devices is only allowed if your site is on HTTPS (SSL secured).