The present article reviews the recent developments in the physics of quantum hydrodynamics and quantum turbulence in superfluid helium and atomic Bose-Einstein condensates. Quantum turbulence (QT) was discovered in superfluid ⁴He in the 1950s, and the research has tended toward a new direction since the mid 1990s. QT is comprised of quantized vortices that are definite topological defects, being expected to yield a model of turbulence that is much simpler than the classical model. The general introduction of the issue and a brief review on classical turbulence are followed by a description of the dynamics of quantized vortices. The first modern topic is a vortex lattice formation in a rotating Bose-Einstein condensate. The second describes the modern trends on quantum turbulence in superfluid helium, addressing the energy spectrum of QT, the possible dissipation mechanism at very low temperatures, temperature-dependent transition to QT, QT created by vibrating structures, and visualization of QT. The last topic is QT in atomic Bose-Einstein condensates.