The dynamics of vesicle shape transformations are governed by the competition between curvature energy, geometrical constraints and viscous dissipation in the surrounding liquid. After a discussion of the general principles, three examples illustrate these concepts. (i) Slow conformal diffusion of vesicles of higher genus arises from the invariance of the curvature elasticity with respect to conformal transformations, (ii) spinodal fluctuations indicate the onset of the budding instability, and (iii) the dynamic pearling instability of cylindrical vesicles is caused by the action of laser tweezers.