The borderlands between elasticity and hydrodynamics lead naturally to a number of moving boundary problems in elastohydrodynamics. I will discuss some phenomena in this rich area involving extreme geometries: the flutter of a slender flag in a breeze (and its relation to fish swimming), the lift on a soft fluid-lubricated solid sliding/rolling near a wall (and its relation to joint lubrication), and the dynamics of fluid-filled tissues (and its relation to rapid movements in some plants).