To turn off data roaming follow this instruction path:
    Settings > under Wireless and network select More...
      (A prerequisite to proceed is that Flight mode is deselected. (Flight/Airplane mode toggles all communication.)
         > Mobile Networks > then deselect "Data roaming"
Data roaming should always be left deselected, unless your device has been unlocked from your cell provider, or you are rich enough to buy a suitable roaming plan from your existing cell provider.
Now that you're data roaming safe, read on for how to further control mobile data usage.
Anytime you don't want mobile data (roaming or not) at the same path as above deselect "Use packet data".
There is the possibility that yourself or a process could toggle packet data on it's own agenda. So to avoid a data leak while roaming. you should have deselected "Data roaming" before turning off "Mobile data".
The "Use packet data" is also user toggled by "Mobile data" on the pop up notifications menu.