The focus of this paper is the European Union and its beginnings. The paper presents a chronological description of the emergence of the European Monetary Union and the official common currency of the euro. The Euro is used by 19 of the 28 European Union countries as a single European currency, and is used by almost 340 million people every day, making this currency the second most common currency in the world. The emphasis is on the idea of introducing the euro, the look and the symbols that each banknote of this common currency possesses. Since each country has to meet the convergence criteria before entering the Eurozone, as defined by the Maastrich Threaty, each criterion has been analyzed in detail below. Also examined are the criteria which Croatia does not currently fulfill the present situation in the Republic of Croatia for each criterion; the criterion of price stability, the criterion of government finances, the criterion of the stability of the exchange rate and the criterion of long – term interest rates. Given the historical similarity between Estonia and the Republic of Croatia with regards to the formation of these states and the years that passed from gaining their independence, Estonia`s entry into the eurozone has been observed. Since there is a public debate in Croatia about whether or not to introduce the euro, the aim of this paper is to present and describe in a simple and concise manner the role of the exchange rate mechanism and the convergence criteria that every country aspiring to enter the eurozone has to meet.