Regarding annuity fees there are two systems in the EPC.
First of all, when missing the initial renewal fee due date, you will have an additional 6 months to validly pay the fee with a surcharge (which is 50% of the fee due).
If you still have not paid the annuity fee within this time frame, the application is deemed to be withdrawn and the European Patent Office will send you a so-called 'Notice of loss of rights' (Rule 112 EPC). You will then have the possibility to ask for re-establishment (Art. 122 EPC). For this you should file a request within two months from said notice*, pay the re-establishment fee and pay the fee that you omitted to pay (+surcharge). Further, you have to provide a statement why you did not pay despite of exercising all due care. This last criterion is quite stringent: you have to show that you have a good working system in place and that the omittance of the payment occurred as an isolated error. There are tons of case law on how to interpret this 'all due care'.
The EPC, however, only relates to European patent applications. After grant, the European patent falls apart into a bundle of national patents and you are obliged to pay the renewal fee to the patent office of each country where your paten has been validate (this is a separate requiremnt after grant). As far as I know in each of the major countries in Europe a similar system of re-establishment is in place, although the requirements and the interpretation of 'all due care' varies from country to country. If you want to know more on a specific country, you should consult a local agent.
*The two months term may be extended if there is a reason that you can not pay (e.g. bankruptcy). Then the term will be 2 months after 'removal of the case of non-compliance' (thus e.g. when the bankruptcy was resolved). However, the total term may not be longer than 1 year after the missed event/payment (in the case of renewal fees the date on which cou still validly pay with surcharge).