To make a claim for lost luggage, start by notifying the airline immediately at the airport's baggage claim office. Fill out a lost luggage report with as much detail as possible. Keep your baggage claim ticket and collect contact information for the airline's customer service. Be patient, as the process may take some time. Keep all documentation, like boarding passes and receipts, for a smooth claim process. Airlines have guidelines for compensation, which may vary. It's essential to stay in contact with the airline throughout the claim process to ensure a successful resolution.