As outlined in our article on New Zealanders and Permanent Australian residency, in general it is not necessary for New Zealand passport holders to apply for a visa prior to coming to Australia. On arrival, they will be granted a Special Category Visa (Subclass 444).
If you have been convicted of a criminal offence (anywhere in the world), this might inhibit your ability to be granted a Special Category Visa on arrival, as the delegate may consider that you do not meet the ‘Character Test’. In general, if a person has served time in prison, or been convicted of certain offences deemed serious, they are more likely to fail the character test.
There are also circumstances under which the delegate must refuse to grant a visa, including where a person has spent more than 12 months in prison.
If you have tuberculosis, this may jeopardise your plans to come to Australia.
In either case some exemptions may be granted, but it is essential to inform the Department of your circumstances in advance, so you know you won’t be turned around at the border.