This paper reports the free longitudinal vibration of axially functionally graded (AFG) tapered nanorods with variable cross-section based on the nonlocal elasticity theory. Elasticity modulus and mass density of the nanorod vary continuously in the axial direction of the nanorod according to a power-law form. Clamped-clamped (C-C) and clamped-free (C-F) boundary conditions are considered. Galerkin method is utilized to obtain free vibration frequencies. The effects of nonlocal parameter, different material composition, taper ratio, different change of the cross-sectional area and the boundary conditions on the free vibration characteristics of AFG nanorod are discussed. For comparison purposes, free vibration frequencies of isotropic rod with sinusoidally varied cross-section and those of tapered nanorod with linearly varied radii are obtained and compared with previously published studies. Good agreement is observed. It is found that the vibration frequencies are overestimated by the classical (local) rod model because of ignoring the nonlocal effect. Also, the nonlocal effects are found to be more pronounced for C-C nanorod end conditions according to C-F nanorod. (c) 2012 Elsevier B.V. All rights reserved.