The basic and standard principle of all Intellectual Property rights including trademarks is that it is territorial, and rights are valid only in the country that grants them. There are only few exceptions, when with a single registration you can have the rights granted in more that one jurisdiction.
The EU trademark is one of such exceptions. Once registered at the European Union Intellectual Property Office the trademark is valid in all EU Member States. Other examples when with a single registration the trademark is protected in more that one country are the African Regional Intellectual Property Organization (ARIPO) and Organisation Africaine de la Propriété Intellectuelle (OAPI).
If there is a need to use a trademark abroad, company needs to apply for a trademark registration in every country. A genuine international trademark does not exist. The international trademark registration system (Madrid System) only simplifies the process for filing multiple national applications.
The international trademark application is filed at World Intellectual Property Organization (WIPO). However, WIPO only conducts a formal examination of your international application. Once approved, your mark is recorded in the International Register and published in the WIPO Gazette of International Marks. WIPO will then send you a certificate of your international registration and notify the IP Offices in all the territories where you wish to have your mark protected. The IP Offices of the territories where you want to protect your mark will make a decision if to allow the trademark being registered or not. So the final word is still said by the national Offices and the trademark can be registered only if in confirms the requirements of the national laws.