The hydrodynamics of swimming microorganisms explores the biophysical and mechanical principles governing locomotion at small scales, where viscous forces dominate and inertia is negligible. Microorganisms, such as bacteria, sperm, and protozoa, use specialized structures like flagella or cilia to move through viscous fluids. Unlike macroscopic swimmers, microorganisms cannot rely on momentum transfer to the fluid, as viscous damping prevents efficient propulsion. Instead, they exploit drag forces through complex, time-dependent deformations of their bodies. At low Reynolds numbers, the flow is governed by Stokes equations, which are linear and time-independent. This leads to unique properties such as kinematic reversibility and the scallop theorem, which states that reciprocal deformations cannot result in net motion. Swimming strategies must therefore involve non-reciprocal body motions to generate propulsion. For example, a flagellum's whip-like motion creates a net force due to drag anisotropy, while cilia generate propulsion through coordinated beating patterns. Key concepts include the resistance matrix for solid bodies, flow singularities, and the role of hydrodynamic interactions. Theoretical frameworks, such as resistive-force theory and slender-body theory, help model flagellar locomotion. Recent research focuses on hydrodynamic interactions, locomotion in complex fluids, and the design of artificial swimmers. These studies highlight the importance of understanding the fundamental physics of low-Reynolds number flows to advance both biological and engineering applications.The hydrodynamics of swimming microorganisms explores the biophysical and mechanical principles governing locomotion at small scales, where viscous forces dominate and inertia is negligible. Microorganisms, such as bacteria, sperm, and protozoa, use specialized structures like flagella or cilia to move through viscous fluids. Unlike macroscopic swimmers, microorganisms cannot rely on momentum transfer to the fluid, as viscous damping prevents efficient propulsion. Instead, they exploit drag forces through complex, time-dependent deformations of their bodies. At low Reynolds numbers, the flow is governed by Stokes equations, which are linear and time-independent. This leads to unique properties such as kinematic reversibility and the scallop theorem, which states that reciprocal deformations cannot result in net motion. Swimming strategies must therefore involve non-reciprocal body motions to generate propulsion. For example, a flagellum's whip-like motion creates a net force due to drag anisotropy, while cilia generate propulsion through coordinated beating patterns. Key concepts include the resistance matrix for solid bodies, flow singularities, and the role of hydrodynamic interactions. Theoretical frameworks, such as resistive-force theory and slender-body theory, help model flagellar locomotion. Recent research focuses on hydrodynamic interactions, locomotion in complex fluids, and the design of artificial swimmers. These studies highlight the importance of understanding the fundamental physics of low-Reynolds number flows to advance both biological and engineering applications.