This paper investigates the bivariate dependence structure for three pairs of exchange rates measured against the US dollar: Euro and Japanese yen (JY), Euro and GBP, Euro and Swiss franc (CHF), in the pre-, post-Euro and the transition periods with the sample period ranging from January 1994 to November 2007. The Deutsche mark (DM) is used for the pre-Euro period. The novelty of the paper is that it employs non-parametric plots, which were derived based on the concept of copula, and a robust semi-parametric method to estimate copula models. The results indicate the changes in the dependence structure from the pre-Euro to the post-Euro period for the pairs DM (Euro)-JY, and DM (Euro)-GBP, with major changes occurring during the initial years of the launch of the Euro. For these two pairs of exchange rates, the model captures asymmetric tail dependence, implying different degrees of co-movements during appreciations and depreciations against the USD. The dependence between the Euro and the CHF remains unchanged, both in strength and structure, over the whole sample period, reflecting a marked tendency of the CHF to follow the fluctuations of the Euro against USD. There has been a tendency for the dependence between Euro and GBP to increase across the whole sample period. The results may be of interest for central banks, international trade, international portfolio diversification and currency risk management.