In this talk I will focus on the physics of Binary Neutron Stars merger obtained from numerical simulations of the Einstein equations coupled to matter. The results are obtained using a semi-realistic descriptions of the equation of state (EOS), where the EOS is described by a seven-segment piece-wise polytropic and a thermal component. One of the important characteristics of the present investigation is that it is entirely performed using only publicly available open source software, the Einstein Toolkit for the dynamical evolution and the LORENE code for the generation of the initial models. After the gravitational-wave event GW150914, observed by the LIGO/Virgo collaboration, the new eve of Gravitational Wave physics has just began and it is clear that accurate modelling of the gravitational wave signal emitted by compact binary sources will play a prominent role. In particular I will present results for the gravitational wave-signal obtained from three-dimensional numerical simulations of the dynamics of binary neutron star (BNS) mergers from the late stage of the inspiral process up to about 20 ms after the system has merged, either to form a hyper-massive neutron star (NS) or a rotating black hole (BH).