In order for you to initiate the eviction process, Arizona law requires certain procedures regarding service of notice of the termination of their lease. If these are done incorrectly (or not at all) and the tenant raises it as a defense, the Court will dismiss the your complaint due to technical defect and you will need to start over.
33-341. Termination of tenancies
A. A tenancy from year to year terminates at the end of each year unless written permission is given to remain for a longer period. The permission shall specify the time the tenant may remain, and upon termination of such time the tenancy expires.
B. A lease from month to month may be terminated by the landlord giving at least ten days notice thereof. In case of nonpayment of rent notice is not required (emphasis added).
C. A tenant from month to month shall give ten days notice, and a tenant on a semimonthly basis shall give five days notice, of his intention to terminate possession of the premises. Failure to give the notice renders the tenant liable for the rent for the ensuing ten days.
D. When a tenancy is for a certain period under verbal or written agreement, and the time expires, the tenant shall surrender possession. Notice to quit or demand of possession is not then necessary.
There are other forms of notices not associated with non-payment of rent. The most common forms are:
3-Day Notice to Cure Breach of Covenant or Quit
3-Day Notice to Quit for Committing a Legal Nuisance