Australia passport holders are visa exempt to enter Luxembourg under the Schengen visa scheme. You can stay up to 90 days within a period of 180 days (6 months), after which you will need to leave the Schengen country you are in. You will be eligible to enter the Schengen area again once you have spent more then 6 months out of these countries. Both the entrance and the exit can be done from different Schengen countries.
Australia passport holders are visa exempt for up to 90 days provided the traveller:
hold a valid passport with a validity of at least 6 months with minimum 2 blank visa pages
have sufficient funds for the trip
have proof of onward/return flights
have all necessary documents for the purpose of entry
IMPORTANT NOTE: It is always recommended that traveller confirm with the airline that they can board without a visa. The final decision is subject to the Immigration Officer's discretion at point of entry.