Push and Fetch Have Stopped Working

Does anyone know how to fix this problem? The only way I get anything is to open the mail app. When I do, I get an avalanche of notification alerts. Everything has worked perfectly until today. I have several different e-mail accounts and the problem affects all of them.

This is happening on both my iPhone and iPad.

 

This happened to me a few times too. No idea why, but what worked for me was to go into settings/mail, and deactivate my accounts one at a time. Deactivate, not delete! After that, open the mail app, and you should be presented with the opportunity to set up a new mail account. At that point, close the app, go back into settings, and reactivate your accounts.

I'm sorry if that isn't quite the right process. There is at least one thread about this on macrumors, and perhaps a search will turn it up.