Recently, two-dimensional transition-metal borides (MBenes) have attracted substantial interest due to their promising properties for electrocatalytic applications. Here, we explore their potential as novel two-dimensional superconductors and topological materials through first-principles calculations on both pristine and surface-functionalized hexagonal MBenes. We conduct a thorough examination of the structural, electronic, phononic, superconducting, and topological properties of 36 compounds with formulas M₂B₂ and M₂B₂T₂ (M = Sc, Ti, V, Zr, Nb, Hf, Ta, Mo, W; T = F, O, OH). Our analysis identifies 21 superconducting MBenes, including four with critical temperatures (T_c) exceeding 10 K, with Ti₂B₂O₂ exhibiting the highest predicted T_c of 24 K based on the McMillan formalism. For the two most promising compounds, Ti₂B₂O₂ and V₂B₂(OH)₂, we further solve the anisotropic Migdal-Eliashberg equations, obtaining zero-temperature superconducting gaps of 6.1 and 3.6 meV and anisotropic T_c values of approximately 32 and 25 K, respectively. Symmetry-indicator-based analysis further reveals nontrivial normal-state band topology in several superconducting MBenes. These results indicate the coexistence of phonon-mediated superconductivity and nontrivial normal-state band topology within this material family, making MBenes promising platforms for future investigations of the possible emergence of topological superconductivity.