Closed-form solutions for the local and nonlocal buckling stretches and loads of Mooney-Rivlin rods for various (i) support conditions, (ii) values of the nonlocal parameter corresponding to (a) the assumption of linearly elastic material, (b) the nonlinear constitutive law of the Mooney-Rivlin material with first order and third order shear deformation theories for rubber-like materials are presented. A perturbation method is used for the solution corresponding to the third order shear deformation theory. Some of the solutions obtained in the current study are compared with those available in the relevant literature. Although the assumption of the linearly elastic behaviour of the material yields perfect local buckling stretch values interestingly (if the limitation on the magnitudes of strains were disregarded), it yields meaningless values for the local buckling loads; basing on the three-dimensional elasticity solution for the Mooney-Rivlin rods. Appealing results are obtained via the nonlinear constitutive law of the Mooney-Rivlin material.