The nature of quasiparticles in d-wave superconductors tends to encourage the quantum-mechanical aspects of vortices. Short coherence length makes the vortex cores small, while presence of gapless fermionic excitations gives rise to certain universal quantum effects. As a result, vortices could behave as quantum particles in very clean d-wave superconductors at low temperatures (unlike the conventional superconductors where vortices are best described as large classical objects). Interesting consequences include a small finite renormalization of vortex mass by the nodal quasiparticles, and absence of Bardeen-Stephen damping of vortex motion in the limits of zero temperature and vanishing core size. These phenomena may have important implications for the "normal" state of the cuprates. In addition, being liberated from strong friction, quantum fluctuations of localized vortices can significantly affect quasiparticle spectra. The local electronic density of states (LDOS) near a quantum fluctuating vortex shows no zero-energy peak, but has sattelite features at energies set by the vortex trapping potential. These are proposed to be the origin of the sub-gap LDOS peaks observed in recent STM experiments near the vortex cores.