There are two (2) types of Landlord/Tenant Proceedings:

Nonpayment Proceeding: Commenced by a Landlord if a Tenant is not paying the monthly rent as required by their lease. The tenant must be in possession of the Apartment in order for the Landlord to commence a summary nonpayment proceeding. If the Tenant is no longer the tenant, and has moved out, the Landlord must   commence what is called a plenary action.  The Housing Court does not have jurisdiction.

Holdover Proceeding: Commenced by a Landlord if a Tenant has no lease or is violating specific terms of the lease (ex. Subletting without permission of the Landlord; Causing a Nuisance; Violating a No-Pet Clause in the lease).

Both types of Proceedings have “Pre-Notice Requirements” (notices that must be served to the tenant before it can be bought to Court). After the Notices are properly served, and the time requirements have expired, then a Notice of Petition and Petition are served upon the tenants. The Notice of Petition / Petition are the equivalent of what is commonly known as the Summons and Complaint in other legal proceedings.

The majority of New York City is apartment living, and as a result there are many legal issues Landlords and Tenants encounter every day.  This is why there is an entire body of law and a separate part of the New York City Civil Court devoted entirely to Landlord / Tenant issues and disputes.