Under Part 101 rules, you are not permitted to fly in Low Flying Zones.
Several other types of airspace are also effectively no-fly zones unless you do certain things:
Within 4 km of an Aerodrome. You are not permitted to fly non-shielded and without a physical barrier between the drone and the aerodrome without consent from the aerodrome operator. You will also need to have completed a NZ approved drone pilot course, and in the case of uncontrolled aerodromes, have an observer with you to look out for other aircraft.
Control Zones. These are near large airports with regular passenger services. You may be surprised how far away from the airport this airspace can be. You are not permitted to fly non-shielded without consent from Air Traffic Control. You can request this with minimum 24 hours-notice via the AirShare app(external link).
Restricted Areas. These are generally a wildlife conservation areas, but can be for other reasons too. You are not permitted to fly without consent from the administering organisation that controls the area.
Military Operating Areas. You are not permitted to fly without consent from the military organisation that controls the area.
The best way to check airspace and who you need to ask consent from is from the AirShare map(external link).
You are permitted to fly in Danger Areas, but you must take into account the hazard present. This is most commonly firearms, demolition, model aircraft and gas efflux.
Learn more about the rules - Intro to Part 101 rules for unmanned aircraft