EU GDPR | Art. 27

Representatives of controller or processors not established in the EU

Article 27
  1. Where Article 3(2) applies (EU Data Subjects data being processed by a non-EU entity), the controller or the processor shall designate in writing a representative in the Union.
  2. The obligation laid down in paragraph 1 of this Article shall not apply to
    1. processing which is occasional, does not include, on a large scale, processing of special categories of data as referred to in Article 9(1) or processing of personal data relating to criminal convictions and offences referred to in Article 10, and is unlikely to result in a risk to the rights and freedoms of natural persons, taking into account the nature, context, scope and purposes of the processing; or
    2. a public authority or body.
  3. The representative shall be established in one of the Member States where the data subjects, whose personal data are processed in relation to the offering of goods or services to them, or whose behaviour is monitored, are.
  4. The representative shall be mandated by the controller or processor to be addressed in addition to or instead of the controller or the processor by, in particular, supervisory authorities and data subjects, on all issues related to processing, for the purposes of ensuring compliance with this Regulation.
  5. The designation of a representative by the controller or processor shall be without prejudice to legal actions which could be initiated against the controller or the processor themselves.

This will probably seen by some as the most contentious article and relates to any entity processing personal data either as controller or processor, where they are not incorporated within the EU.  In effect the legislation states that they should employ the services of a representative to act on their behalf, such representative entity needs to be incorporated within the EU.

Note paragraph 5 and that it does not negate the responsibility of the controller or processor.