Summary: remember to turn off all roaming - not just data-roaming - to avoid excessive phone company charges.
When traveling with your phone, everyone now seems to be aware of the excessive and unexpected costs that show up on the phone bill after your travel.
Phone companies very sneakily gouge customers on roaming charges. So when you search for this topic on the web, there are numerous pages that explain how to turn off roaming.
That help is misleading, since it is for data roaming only, and that is not sufficient. Data is the big problem of course, but there is also the problem of voice and text - sending and receiving phone calls and text messages.
And at least on the Android phones, there is no indication of this happening too. You take your phone from US to say Canada or Europe, and think you are all good - you've turned off data roaming, which was easy to do in the Settings. But then your phone rings - you are surprised, hey, US phone works in this country? You pick up the phone, you talk, and BAM! you are now liable for multiple dollars per minute of charges. And there is no help in the Android menus to indicate you are being billed roaming charges. You did not receive any warning, and if you continue to use your phone and make calls, you'll get hit with large bills when you get back home.
Sadly, there is no easy way to turn off voice and text roaming on a Android smartphone.
This is too bad - it lays a trap for unwary customers.
A multi-step workaround: Go into Settings, More... button, and turn on Airplane mode.
But this turns off WiFi too, which is not desirable. Go back into Settings and turn on WiFi.
This is better! Took a few steps, but now you won't have any phone company surprises.
And GPS still continues to work, at least in one of the phones I tried.