The equilibrium relations for water and sediment transport refer to the relative balance of sediment transport and the relative stability of river courses formed by the automatic adjustment of riverbeds.This is the theoretical basis for the comprehensive management of sediment in the Yellow River.Based on the theories of sediment carrying capacity and the delayed response of riverbed evolution,in this study,the equilibrium relations for water and sediment transport in the Yellow River are established.These relations include the equilibrium relationships between water and sediment transport and bankfull discharge in the upper and lower Yellow River and between water and sediment transport and the Tongguan elevation in the middle Yellow River.The results reveal that for the Ningmeng reach,the Tongguan reach,and the lower Yellow River,erosion and deposition in the riverbeds are adjusted automatically,and water and sediment transport can form highly constrained equilibrium relationships.These newly established equilibrium relationships can be applied to calculate the optimal spatial allocation scheme for sediment in the Yellow River.