We investigate the importance of quantum orbital fluctuations in the orthorhombic and monoclinic phases of the Mott insulators LaVO3 and YVO3. First, we construct ab-initio material-specific t2g Hubbard models. Then, by using dynamical mean-field theory, we calculate the spectral matrix as a function of temperature. Our Hubbard bands and Mott gaps are in very good agreement with spectroscopy. We show that in orthorhombic LaVO3, quantum orbital fluctuations are strong and that they are suppressed only in the monoclinic 140 K phase. In YVO3 the suppression happens already at 300 K. We show that Jahn-Teller and GdFeO3-type distortions are both crucial in determining the type of orbital and magnetic order in the low temperature phases. The Mott insulating t 2 2g perovskites LaVO 3 and YVO 3 exhibit an unusual series of structural and magnetic phase transitions ( Fig. 1) with temperature-induced magnetization reversal phenomena [1] and other exotic properties [2,3]. While it is now recognized that the V-t 2g orbital degrees of freedom and the strong Coulomb repulsion are the key ingredients, it is still controversial whether classical (orbital order) [1,4,5,6,7,8] or quantum (orbital fluctuations) [2, 9] effects are responsible for the rich physics of these vanadates.At 300 K, LaVO 3 and YVO 3 are orthorhombic paramagnetic Mott insulators. Their structure (Fig. 2) can be derived from the cubic perovskite ABO 3 , with A=La,Y and B=V, by tilting the VO 6 octahedra in alternating directions around the b-axis and rotating them around the c-axis. This GdFeO 3 -type distortion is driven by AO covalency which pulls a given O atom closer to one of its four nearest A-neighbors [10,11]. Since the Y 4d level is closer to the O 2p level than the La 5d level, the AO covalency increases when going from LaVO 3 to YVO 3 and, hence, the shortest AO distance decreases from being 14 to being 20 % shorter than the average, while the angle of tilt increases from 12 to 18 0 , and that of rotation from 7 to 13 0 [12,13]. Finally, the A-cube is deformed such that one or two of the ABA body-diagonals is smaller than the average by, respectively, 4 and 10 % in LaVO 3 and YVO 3 . These 300 K structures are determined mainly by the strong covalent interactions between O 2p and the empty B e g and A d orbitals, hardly by the weak interactions involving B t 2g orbitals, and are thus very similar to the structures of the t 1 2g La and Y titanates [10,11].