Abstract : Constitutive equations of finite nonlocal elasticity are obtained. Thermodynamic restriction are studied. The linear theory is given for anisotropic and isotropic solids. The physical and mathematical properties of the nonlocal elastic moduli are explored through lattice dynamics and dispersive wave propagations. The theory is applied to the problems of surface waves, screw dislocation and a crack. Excellent agreements with the results known in atomic lattice dynamics and experiments display the power and potential of the theory.