We propose robust and distributionally robust formulations of the risk budgeting problem under the mean absolute deviation (MAD) risk measure. The proposed models address uncertainty in asset returns and their distribution while allocating portfolio risk according to prescribed risk budgets. We first develop a robust model in which historical return observations are perturbed within a correlated budget uncertainty set. Portfolio deviations are measured relative to the mean of the same perturbed empirical distribution, accounting for the effect of uncertainty on both returns and their mean. We then introduce a distributionally robust model in which the return distribution is allowed to vary within a Wasserstein ambiguity set centered at the empirical distribution, and portfolios are constructed by considering the worst-case MAD over the admissible distributions. Tractable optimization formulations are derived for both models. Numerical experiments using real-world stock market data compare the proposed approaches with nominal risk budgeting, minimum MAD, and standard-deviation risk budgeting models. The results suggest that accounting for uncertainty can improve portfolio stability and out-of-sample performance. In particular, the proposed models achieve the lowest turnover and the highest average out-of-sample expected return, Sharpe ratio, and Calmar ratio, together with the smallest average deviations from the risk contributions.