In this paper, we consider the Cauchy problem for the 3D MHD system with nonlinear damping terms κ|u|α−1u and γ|b|β−1b and prove the existence and uniqueness of global strong solution when one of the following two conditions is verified (1)3≤α<4,β≥6α−1+1;(2)α≥4,β≥1. This greatly improves the previous results. Moreover, we also discuss the uniqueness of weak solutions when α and β satisfy certain conditions. Finally, we study the decay of weak solutions when α,β>73 and obtain the upper bound for the derivative of strong solution when 3≤α<4,β≥6α−1+1 or α≥4,β>73.