To claim compensation for a delayed Delta Airlines flight, begin by understanding your passenger rights and eligibility. Gather essential documents such as boarding passes and tickets. Contact Delta customer service to inquire about their compensation policies. If eligible, file a claim via their website or other designated channels, detailing the delay's specifics and inconveniences faced. Maintain records of all communications. Exercise patience during the claim processing period, escalating if necessary. Consider utilizing air passenger rights organizations for guidance. As a last resort, legal action can be pursued. Be mindful of specific time limits for filing claims and variations in compensation policies.