For some time, Teamviewer has had an app that allows one to control a remote PC from an Android device.
They have now introduced a new mobile app called Teamviewer QuickSupport
that allows a user to control certain Android devices (Samsung, Sony and a few others) from a PC.
The app is remarkably flexible, allowing not just remote control of the device, but file transfers in both directions through a 2-pane file explorer, screen captures and remote process control, among other features. Some quick testing on my Nexus 10 (Samsung) showed that everything worked smoothly.
They also have a much less powerful version for iPad/iPhone, which I haven't tried, since I have no iOS devices.