We investigate neutron star moments of inertia from Bayesian posterior probability distributions of the nuclear equation of state that incorporate information from microscopic many-body theory and empirical data of finite nuclei . We focus on PSR J0737-3039A and predict that for this 1.338 M _ { \odot } neutron star the moment of inertia lies in the range 1.04 \times 10 ^ { 45 } g cm ^ { 2 } < I < 1.51 \times 10 ^ { 45 } g cm ^ { 2 } at the 95 % credibility level , while the most probable value for the moment of inertia is \tilde { I } = 1.36 \times 10 ^ { 45 } g cm ^ { 2 } . Assuming a measurement of the PSR J0737-3039A moment of inertia to 10 % precision , we study the implications for neutron star radii and tidal deformabilities . We also determine the crustal component of the moment of inertia and find that for typical neutron star masses 1.3 M _ { \odot } < M < 1.5 M _ { \odot } the crust contributes 1 - 6 % of the total moment of inertia , below what is needed to explain large pulsar glitches in the scenario of strong neutron entrainment .