Airline’s liability on lost or damaged luggage
Occasionally, a suitcase falls off a ramp and gets run over by a forklift or simply gets lost. You'll know right away if something like that has happened to one of your bags.

 Damaged luggage
If the airline has manhandled your luggage, you can seek reimbursement. If the bag is ripped or a wheel is missing, you may even be able to get it fixed while you wait (several major airports, including Minneapolis/St. Paul and Atlanta Harts field, have luggage-repair shops on site). Some airlines, including United, have replacement bags they'll give you on the spot at major airports.
Don't expect the airline to reimburse you for nicks and scratches, however. Those are considered normal wear and tear. And don't expect reimbursement if a fragile item breaks (an exception is something packed in its original, factory-sealed container, which should be able to withstand rough handling).
When you claim your luggage, check for damage. If a bag is open, look inside to see whether any of your belongings are damaged or missing. If so, immediately report the damage and any missing items to the airline's baggage personnel.
Be sure to get the name of the person with whom you speak. If you are reporting the damage from a location other than the airport, the DOT recommends sending a certified letter to the airline as a follow up.

 Lost luggage
Last but not least, if a carrier loses your luggage, you should go immediately to the baggage or customer service office, which is usually located near the baggage claim area. Fill out the proper report and refer to the packing list made at the beginning of the trip. Receipts for newly purchased luggage, expensive clothes and other items of value help when making claims.
Airline’s baggage liability :
Airlines are not responsible for carry-on luggage. But the airlines are liable if checked baggage is lost or damaged.

Baggage liability for Domestic flights :

• Travel wholly within the 50 United States, Puerto Rico, and the U.S. Virgin Islands.
• Liability is limited to $2,800 per ticketed passenger, unless a higher value (for checked baggage) is declared in advance and additional charges are paid.
• Excess valuation may not be declared on certain types of articles.
• No liability for electronic equipment, photographic equipment, jewelry, cash, computer equipment, or other similar valuable items.

Baggage liability for International flights :

• $9.07 per pound for a 70-pound bag, to a maximum of about $640.

Airlines don't pay the liability amount for every claim. Rather, they estimate the value of the contents of your bag based on the report you file. (It helps if you've kept a list in advance of the contents of your bag.) Depreciation is considered. Airlines are free to deny any claims that seem exaggerated, and they may ask for sales receipts or other documentation to prove value.