British customers reported not being able to use mobile data to access the internet and the operator's network on Thursday after disruption began at about 5am, with full service not returning until early on Friday morning.
On Friday, the telecommunications firm confirmed they would be offering incentives by way of apology.
What are O2 offering customers?
O2 have said that monthly customers, small and medium business customers and mobile broadband customers will be credited with two days of monthly airtime subscription charges by January.
For pay-as-you-go broadband customers, O2 is offering a 10% discount on a bolt on purchase available from early next year.
What is the likelihood of further compensation and how do you pursue it?
Guidance from regulator Ofcom says that, depending on circumstances, it "may be appropriate" for mobile networks to offer some form of compensation for loss of service.
What if you are unhappy with a company's response?
If customers are not happy with how a complaint has been responded to, or it takes more than eight weeks to be resolved, a further complaint can be submitted to an independent Alternative Dispute Resolution scheme.
What about other networks which were affected?
Other mobile networks, including Tesco, Giffgaff and Sky, also experienced problems because they use O2 services.
Sky Mobile has already confirmed it will compensate customers with a day of free unlimited data.
"We're really sorry for the issues you experienced on Sky Mobile yesterday," the network said.
"We're giving you a day of free unlimited UK data this Saturday 8 Dec, so you can catch up on all the things you’ve missed. You don't need to do a thing."