UAV's can be legally operated with Australia, however there are certain rules and regulations you need to be aware of.
You do not require a license if you are flying your UAV for private or hobby purposes. You will require a license if you intend on operating your UAV for commercial reward.
The basic rules for UAV operation in Australia are:
1) You may only operate during the day and in visual conditions, i.e: not in cloud, fog etc.
2) You must maintain at all times visual line of sight control with your UAV.
3) You must keep your UAV a minimum of 30 metres away from other people, vehicles, buildings etc.
4) You must not fly higher than 400ft or 120 metres.
5) You must not fly within 5.5km of an airport or airfield.
6) You must not fly over a populous area, which includes beaches, houses, populated parks, sports ovals (when a game is in session).
CASA is the governing body of Civil Aviation in Australia, they have a page dedicated to UAV's and the associated rules, that page can be found here: https://www.casa.gov.au/operations/standard-page/remotely-piloted-aircraft-rpa
They also provide a handy PDF, which provides the rules listed above in a easy to read summary, https://www.casa.gov.au/sites/g/files/net351/f/_assets/main/lib100071/flying_with_control_model.pdf
If you intend to operate your UAV for commercial reward, you will need to obtain a license and develop UOC (UAV Operators Certificate). Some holders of UOC's may consider letting you operate under their UOC with an agreement in place.
CASA's official guidance can be found here https://www.casa.gov.au/operations/standard-page/how-become-safe-rpa-operator