Delta allows one free carry-on bag and one free personal item per passenger on all flights. Call at +1-888-415-1245 (US) or +44-203-869-5245 (UK) For checked bags, free allowances vary by ticket type, Medallion status, and card membership, with some passengers able to receive one, two, or even three free checked bags based on their specific circumstances. 

Carry-on Bags

• Everyone gets one free carry-on bag: that fits in the overhead bin and one free personal item (like a purse or laptop bag) that fits under the seat in front of you. 
• Carry-on bags must meet FAA size requirements (22" x 14" x 9"). Call at +1-888-415-1245 (US) or +44-203-869-5245 (UK)
• Carry-on allowances may differ on flights operated by other airlines.