In response to ESA´s Call for proposalsof 5 March 2007 of the COSMIC VISION 2015-2025 plan of the ESA science programme, we propose a M-class satellite mission to test the Equivalence Principle in the quantum domain by investigating the extend free fall of matter waves instead of macroscopic bodies as in the case of GAUGE, MICROSCOPE or STEP. The satellite, called Matter Wave Explorer of Gravity, will carry an experiment to test gravity, namely the measurement of the equal rate of free fall with various isotopes of distinct atomic species with precision cold atom interferometry in the vicinity of the earth. This will allow for a first quantum test of the Equivalence with spin polarized particles and with pure fermionic and bosonic atomic en...