First of all, in this paper, the topology and operation of the three-phase three-level Z-source inverter based on neutral-point-clamped (Z-NPC) are studied. Moreover, different combinations of permissible switching states and control signals are explained for this inverter. In this paper, the topology of the three-phase three-level Z-NPC inverter is extended for an n-level state. Also, a combination of allowed switching states with relevant mathematical equations is presented for the proposed n-level Z-NPC inverter. In comparison with multilevel voltage-source inverters (only voltage-boost capability), the proposed multilevel Z-NPC inverter is a single-stage converter and it has a buck-boost capability of voltage. On the other hand, the control of two-stage converters compared to single-stage converters can be more difficult because of existing more active and passive components. In this paper, two new PWM control methods are also proposed for various multilevel Z-NPC inverters. One advantage of the proposed PWM control methods in comparison with conventional PWM control methods is maintaining the charge balance of the dc-link capacitors in neutral point. The correct performance of the proposed multilevel Z-NPC topology and PWM control methods are verified by the obtained results of analysis and simulations performed in the PSCAD software.