Target crossbows tend to be bespoke devices and must include a recurve prod (the bow).
Sporting crossbows are the commerical style crossbows available in either recurve or compound.
In many States, the ownership and use of crossbows is either banned or restricted (requiring a licence or permit). You should check the requirements in your State with your RGB (Regional Association).
People who shoot crossbows are called "ARBALISTS".
Archery Australia caters for Crossbows within its rules. National and State competitions are conducted periodically to include crossbow divisions. Annually, through the Crossbow Committee, National Championships are held.
Archery Australia has a Crossbow Committee to represent the interests of arbalists in Australia.
Contact the Crossbow Committee Chair: email@example.com