I have Italian citizenship and I would like to go work in Australia or New Zealand for an undefinite amount of time.
I don't have a bachelor's degree, would it be enough for me to get a job offer as a programmer to then get a working visa?
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The best way for you to get started living in Australia or New Zealand is probably to get a working holiday visa (assuming you're under 30).
There are restrictions, such as the maximum term with a single employer is 6 months (Australia) or 3 months (New Zealand). After you get some local work experience, you may be able to get a proper job offer, perhaps from one of the employers you worked for during your working holiday. Then, with a job offer, you can apply for a longer term working visa. If that all works out, then you may be able to turn that working visa into a permanent residence visa.