There are different reasons for employment relationship to come to an end such as resignation, dismissal or end of fixed-term Contract (CDD). Luxembourg Labour Law as in most countries in European Community more specifically EU has a very precise way to go about it. Please find below link to a comprehensive guide of ending employment contracts in Luxembourg:

http://www.expatica.com/lu/employment/Labour-law-in-Luxembourg-Ending-employment_104491.html.