The Dublin-based carrier said it made the decision because flights are being delayed because of a lack of space in overhead lockers.
From November 1 passengers will only be able to carry a small bag such as a handbag or laptop bag on board, with wheelie bags being placed in the hold free of charge.
Priority boarding customers will continue to be able to keep both bags with them on the aircraft.
Ryanair also announced it is increasing its check-in bag allowance from 15kg to 20kg, and reducing the standard fee from £35 to £25 for a 20kg bag.
The airline’s chief marketing officer, Kenny Jacobs, said he hoped the changes will “eliminate flight delays” caused by not having sufficient overhead cabin space on busy flights.
He went on: “These bag policy changes will cost Ryanair over 50 million euros (£46 million) per annum in reduced checked bag fees.
“However, we believe offering bigger bags at reduced fees will encourage more customers to consider checking in a bag, which will reduce the high volume of customers we have with two carry-on bags at the boarding gates, which is causing flight delays due to large numbers of gate bag and cabin bag offloads.”
The changes will apply to all existing and future bookings.