This paper develops a unified framework for asymptotically minimax robust hypothesis testing under distributional uncertainty, applicable to both Bayesian and Neyman--Pearson formulations (Type-I and Type-II). Uncertainty classes based on the KL-divergence, $α$-divergence, and its symmetrized variant are considered. Using Sion's minimax theorem and Karush-Kuhn-Tucker conditions, the existence and uniqueness of the resulting robust tests are established. The least favorable distributions and corresponding robust likelihood ratio functions are derived in closed parametric forms, enabling computation via systems of nonlinear equations. It is proven that Dabak's approach does not yield an asymptotically minimax robust test. The proposed theory generalizes earlier work by offering a more systematic and comprehensive derivation of robust tests. Numerical simulations confirm the theoretical results and illustrate the behavior of the derived robust tests.