Being a practicing criminal jurist I have chosen the subject matter of my examination paper from the new field of European Union law. Since 2004, Hungary has been a part of the European Union, which is regarded as a unified juridical territory of freedom, security and justice. The European arrest warrant is a quick, effective but humane measure of criminal investigation in this common unified territory. Its concept, necessity and application are outlined briefly in the following. Difficulties caused by the simultaneous application of different national laws of Member States may be overcome by the autonomous concept of the European Union law. It is demonstrated through the example of a cocaine smuggler that the same fact may infringe different rules of national criminal laws in each Member State and can be a subject to different classifications in those states. Therefore according to settled case law of the Court the relevant criteria for assessing the criminal act are the material facts and acts understood as the existence of a set of facts and acts which are inextricably linked together, irrespective of the legal classification given to them or the legal interest protected in national level. In addition some examples of case law of the European Court of Justice are given.