Moodle for mobile
Moodle's two standard themes, Clean and More, are responsive, bootstrap-based themes meaning they adapt well to mobile devices.
An administrator can also set a theme for mobile devices in Administration > Site administration > Appearance > Themes > Theme selector. The Simple theme is multi device optimized. The Standard fluid theme is a good start if you like to code your own responsive web design theme in Moodle, or if you use the 'standard' theme and would like to make it ready for smartphones and tablets.
Notifications may be pushed to a mobile device if the administrator configures this in Site administration>Plugins>Message outputs>Mobile notifications.