Greece is a Member-State of the European Union and has ratified the Schengen Agreement. Citizens traveling inside the E.U. just need to display their police I.D. Card.

However, a passport is necessary for a number of other transactions, such as currency exchange, rent a car, etc. 

Visas are not required by citizens of Member-States of the European Union. Greece follows the provisions of the Schengen Agreement, which abolished controls on common internal lands, at air and sea borders and allows Member-State citizens to travel around without a visa for a short stay about three (3) months.

However, airlines and other carriers require a valid passport and/or police I.D. Card or other form of official identification means. 

Citizens of the majority of the countries that are not Member-States of the Schengen Agreement require a visa to enter Greece and the E.U. Visitors from these countries can obtain further information from the Hellenic Embassies or Consulates in their countries, or even from their travel agencies.