We apply total-variation (TV) regularization methods to Abel inversion tomography. Inversions are performed using the fixed-point iteration method and the regularization parameter is chosen such that the resulting data fidelity approximates the known or assumed statistical character of the noisy data. Five one dimensional examples illustrate the favorable characteristics of TV regularized solutions: noise inversion suppression and density discontinuity preservation. One experimental and two simulated examples from X-ray radiography also illustrate limitations due to a linear projection approximation. TV regularized inversions are shown to be superior to squared gradient regularized inversions for objects with density discontinuities. We also introduce an adaptive TV method that utilizes a modified discrete gradient operator acting only apart from data-determined density discontinuities. This method provides improved density level preservation relative to the basic TV method.