Further to Jim McConalogue's excellent post on EU-wide extradition, consider an alarming example cited recently by Alasdair Palmer:
Fair Trials International, a charity that reports on the standards of justice prevalent in other countries, has investigated the case of a Briton who was on holiday in Romania in 2004 and found himself being charged with having sexual relations with a minor. He was detained in Romania on that charge for three months. His alleged victim could not be traced and did not appear or give evidence at any court hearing. In November 2004, he was released and required to leave Romania within five days - which he did.
In March 2007, he received an email from the British embassy in Romania informing him that the Romanian courts had held a trial which had convicted him in absentia of sexual abuse. The court had sentenced him to seven years in prison - a sentence that had been confirmed on appeal. The Romanian government issued a European Arrest Warrant, and the British man was sent back to Romania. After another trial which lasted less than one hour, and at which the alleged victim was again not present, the seven-year sentence was confirmed. The Briton is now in prison in Romania - where he will stay until he has served his sentence.
That case is not a piece of spurious anti-EU propaganda: it actually happened. Not one of the EU's existing "guarantees" made the slightest difference - and the new law would sweep away even those.