In this talk, I will present our two recent studies on realizing topological properties via Rashba and Dresselhaus spin-orbit coupling (SOC) in exotic quantum materials. First, I will introduce twisted type-II Rashba homobilayers as a new platform for achieving topological moiré flat electronic bands. Using bismuth telluride iodide (BiTeI) as an example, we demonstrate the formation of narrow flat bands and transitions from valley Hall to quantum spin Hall insulators by varying twist angles. The tunability in SOC, interlayer interactions, and twist angles enhances the potential of Rashba materials for observing correlated topological phenomena. In the second study, we extend SOC to emerging Lieb-lattice altermagnetic materials, predicting the axial Hall effect, a Berry-curvature-driven anomalous Hall response. We identify the axial direction as a hidden topological degree of freedom, and first-principles calculations confirm the emergence of this effect in strained altermagnets, particularly in ternary transition-metal dichalcogenides. The axial Hall effect originates from the interplay between Dresselhaus SOC and the intrinsic piezomagnetic response of Lieb-lattice altermagnets. These findings emphasize the critical role of SOC and noncollinear spin textures in altermagnets and open new pathways for exploring intrinsic Hall phenomena in topological magnetic systems.