Recent experimental and computational studies of swimming hydrodynamics have contributed significantly to our understanding of how animals swim, but much remains to be done. Ten questions are presented here as an avenue to discuss some of the arenas in which progress still is needed and as a means of considering the technical approaches to address these questions. 1. What is the three-dimensional structure of propulsive surfaces? 2. How do propulsive surfaces move in three dimensions? 3. What are the hydrodynamic effects of propulsor deformation during locomotion? 4. How are locomotor kinematics and dynamics altered during unsteady conditions? 5. What is the three-dimensional structure of aquatic animal vortex wakes? 6. To what extent are observed propulsor deformations actively controlled? 7. What is the response of the body and fins of moving animals to external perturbations? 8. How can robotic models help us understand locomotor dynamics of organisms? 9. How do propulsive surfaces interact hydrodynamically during natural motions? 10. What new computational approaches are needed to better understand locomotor hydrodynamics? These ten questions point, not exclusively, toward areas in which progress would greatly enhance our understanding of the hydrodynamics of swimming organisms, and in which the application of new technology will allow continued progress toward understanding the interaction between organisms and the aquatic medium in which they live and move.