ARC/Annual
EASA ?
If your ARC has run out-you must get a permit to fly it or you C of A is not valid!
If your ARC is valid and its just out of annual check you might be able to fly it depending on your maintenance programme, if you are on the LAMP ask your engineer they should ok it for up to a month.
CAA CofA ?
Your engineer can arrange conditions A as mentioned before if the CofA has run out.