This thesis investigates the motion and breakup of droplets in low-Reynolds-number flows, focusing on two aspects. In the first part, we study the breakup of droplets in subcritical flow conditions, when there exists a linearly stable solution for the droplet shape, but a finite amplitude perturbation might trigger instabilities. Thus, there exists a finite basin of attraction of the stable solution, whose boundary separates droplets that break from those recovering the stable shape. Our effort is mostly devoted to the exploration of the state space in which the basin boundary is defined. To this end, we proceed by adapting theories initially developed to study laminar-turbulent transition, namely nonmodal analysis and edge tracking. We study the influence of non-normal effects in the breakup of a rising droplet, showing that the optimal shapes found with nonmodal analysis are more efficient in triggering breakup than initially ellipsoids droplets. Afterwards, we investigate the relevance of edge state in the breakup of droplet in uni-axial extensional flows, finding that edge states select the path toward breakup. The exploration of the bifurcation diagram reveals a similar situation for bi-axial extensional flows, where droplets are squeezed along the axis instead of being extended. In the second part we develop a joint chemical-hydrodynamics model to study the motion of bubble-propelled conical microswimmers. We conclude that the chemistry and the hydrodynamics partially decouple. In fact, chemistry dictates the time scale at which the microswimmer moves while the hydrodynamics governs the attained displacement. We furthermore find the geometrical and chemical parameters that optimize the swimming velocity. The effects of bubble deformability are then included. In this case, the swimming velocity is optimal for small cone opening angles. Furthermore, we find that the swimming efficiency, measured in displacement attained per fuel consumption, decreases when the bubble is more deformable. Finally, we study the motion of a sphere inflating close to a wall, which is relevant to the study of conical microswimmers and allows us to revisit the classical settling sphere problem. We find that depending on the boundary conditions imposed on the sphere, whether it is a rigid shell or a perfect free-shear bubble, the sphere-wall gap will close or open in time.