what happens when there is no lease agreement and the tenant wants to move out?

A term of a lease and time periods for notices to terminate is generally determined by the lease itself. You indicate that you have not signed a new lease agreement in 13 years; however you are very likely still under the original lease agreement (which probably had a holdover provision).  Even the absence of a holdover provision, your lease most likely has been renewing month to month.  Determining whether there was a written contractual lease agreement and whether it is still effective will most likely determine the answer to your question regarding the amount of notice (if any) which was required.  However the general answer is that notice should be given at least one month prior to termination for a month to month tenancy.  NRS 118A.210 provides "Unless the rental agreement establishes a definite term, the tenancy is from week to week in the case of a tenant who pays weekly rent and in all other cases the tenancy is from month to month." Usually the amount of notice which you are required to give prior to vacating is defined in the Lease Agreement and (on a month-to-month lease) is a full month. You have inquired about responsibility for damage.  The landlord is able to withhold security deposits and/or charge you for damage which constitutes a default under the lease.  The landlord is not entitled to charge you for normal wear and tear.  The landlord is required to provide the tenant with an itemized written accounting of the disposition of the security deposit no later than 30 days after the termination of the tenancy by handing it to the tenant personally at the place where the rent is paid, or by mailing it to the tenant. The Tenant has a right to dispute the accounting by sending a written response within 30 days after receiving the itemized written accounting. If the landlord fails or refuses to return the remainder of a security deposit within 30 days after the end of a tenancy, the landlord may be liable to the tenant for damages.