The aim of this research is to extend the classical LMM to a multi-curve framework and to analyze the impact of this extended model on the most liquid exotic interest rate derivatives. A possible parametrization for the instantaneous volatility and correlation structure is presented and the (log-)normal dynamics of the OIS rates under different measures are obtained. The forward LIBOR rates are modeled at a constant additive spread over the OIS curve. An analytical closed-form approximation of the European swaption volatility in the multi-curve framework is derived and its accuracy is verified by comparing the Monte Carlo prices of a set of European swaptions with the corresponding prices obtained using the approximation. It is demonstrated that the approximation reaches the highest accuracy for swaptions characterized by short underlying tenors and strikes close to the swap rate. The multi-curve LIBOR Market Model is calibrated to the swaption market applying this approximation. Using the calibrated model distinct Bermudan swaptions are priced by means of Monte Carlo. These prices are compared to the corresponding prices obtained using the one-factor Hull-White model and the impact of the model selection is analyzed.