See when a domain was registered, when it expires, who the registrar is, and any status flags that could take it offline.
An expiry date or a hold status takes a whole domain down, mail included, and it is the first thing worth ruling out when everything stops at once.
A public directory of domain registration details: when it was registered, when it expires, and who the registrar is.
Most registrars apply privacy protection, replacing personal details with a proxy, which is normal and expected.