I need the functionality to request user's attention for my application or a single stage.

In Windows 7 this can currently be done with the Stage.toFront() method, which makes the task bar icon blink.

For Mac OS X there's a functionality, which makes the dock icon bounce (and toFront() just brings the Stage to front without bounce)
See:
http://stackoverflow.com/questions/15079783/how-to-make-my-app-icon-bounce-in-the-mac-dock

For Mac you need a 3rd party library and then call:
com.apple.eawt.Application.getApplication().requestUserAttention(false);


It would be nice, if javafx.application.Application had a method like this:
javafx.application.Application#requestUserAttention(boolean)

which would make the application blink in the task bar for Windows 7 and make it bounce in the dock for Mac OS X. (other platforms needs to be evaluated).

Maybe such a method could also be available for Stage.

Similar methods/functionality from com.apple.eawt.Application could also be evaluated (e.g. populating Dock or Taskbar menu), but this should probably be another feature request.