The short answer: sometimes. Mail alone is almost never sufficient for initial service on a new defendant. But it plays a key supplemental role in substitute service and ongoing case notifications.
You cannot serve a new defendant with just a letter in the mail. Arizona Rule 4.1 requires personal service, substitute service, or a court order for alternative methods.
Protective orders must be personally served. Mail service does not trigger the criminal enforcement provisions of the order.
Subpoenas compelling appearance require personal service. Mailing a subpoena creates no legal obligation to appear.
After leaving documents with a person at the residence or workplace, you must also mail a copy to the defendant's last known address.
Once a party has appeared in a case, subsequent documents (motions, notices) may be served by mail on their attorney of record.
Service on some government entities may be accomplished by certified mail under specific statutory provisions.