The most obvious restriction on carry-on luggage is the size. Carry-on luggage is measured in "linear inches," which means the length plus the width plus the height. For Continental, Delta and United Airlines, the maximum size of carry-on luggage is 45 linear inches. Continental imposes an additional restriction: luggage cannot weigh more than 40 pounds. Other airlines may have different size restrictions; be sure to check with the individual airline in order to make sure your luggage is within its limits.
The Transportation Safety Administration (TSA) has imposed restrictions on what people are not allowed to carry onto flights. A general rule: if an item appears that it can be used to stab, beat or shoot someone, explodes or ignites easily, or has toxic fumes, it probably cannot be brought onto a flight. For a comprehensive list, see the TSA website.
Spirit Airlines is the first airline to charge people for some carry-on bags that, on other airlines, would be free. A bag of 42 linear inches or smaller can be brought onto a flight for free; if the bag is larger than that, Spirit imposes a fee. As of October 2010, this fee is $20.