Many, many hikers use running shoes or trail runners while backpacking.

Why I like trail runners over boots:

1. Much, much, much lighter on my feet. I cannot begin to describe how much better this is at the end of the day.

2. No blisters, ever (assuming they fit properly.) My perfectly fitted big heavy stiff boots gave me monster heel blisters every hike for years and years. Long after they were "broken in."

3. Mesh trail runners dry very quickly after they get wet. My big heavy leather boots would take days to dry, and on some trips they never dried.

4. Better for my ankles. The conventional wisdom that one needs boots for ankle protection turns out to be wrong, at least in my case -- I almost never "roll" my ankle any more, vs. several times per trip wearing boots.

5. More control over foot placement, more precision. I think this is mostly due to the light weight (and probably explains #4.)

I made the switch in April, 2000, and will never go back. I've worn trail runners with loads from 15-55 pounds, all four seasons, in all kinds of weather from blistering heat to blizzards. (I do switch to Goretex trail runners in the winter.) Love 'em.