In foreign countries mine is on my person at all times either in a pants pocket or in an ankle carrier when in dodgy areas. Photos of the passport are on my computer and BlackBerry and photo copies of the picture page are in my hotel safe and luggage.
There are people who are frightful of theft or loss and keep their passport in a hotel room safe and carry photocopies. I believe that is a mistake. The passport is the only internationally recognized form of ID and if you need it you generally really need it and a tattered and creased photocopy might just not do.