I plan to bring a carry-on (duffel-like with wheels) which will go in an overhead bin. Will I be permitted to also take with me a backpack that will fit under the seat in front of me? I would not be bringing anything more.)
Suzanne, it depends on the airline. Some are strict 1 bag only. While some will let you bring 1 carry on + 1 personal item (ie: laptop case, purse). I've been able to bring my 19" carry on + 1 school backpack going to Europe... but on one occassion had to stuff my bag in the case to keep to the 1 bag rule.
Many airlines are starting to enforce a strict weight limit as well. Check with your carrier. Lufthansa, e.g., limits you to about 17 pounds per bag. That's pretty easy to exceed if you're carrying anything other than just clothing. If it weighs more than that, they can force you to check it (which, frankly, is far from the worse thing that could happen).