How do I find out if a training provider is a registered training organisation, whether they are authorised to deliver a particular course, qualification or unit and whether they are regulated by ASQA?

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Training.gov.au
https://training.gov.au/
National register for training in Australia.  It contains information about Registered Training Organisations(RTOs), Nationally Recognised Training (NRT), and the approved scope of each RTO to deliver NRT as required in national and jurisdictional legislation within Australia.

My Skills
https://www.myskills.gov.au/
The My Skills website is the national directory of vocational education and training (VET) organisations and courses. It is an Australian Government initiative to enable consumers to search for, and compare, VET courses and training providers.

CRICOS Register
http://cricos.education.gov.au/
Official government website that lists all Australian education providers that offer courses to people studying in Australia on student visas and the courses offered.