  This brief review summarizes recent theoretical and experimental results
which predict and establish the existence of quantum droplets (QDs), i.e.,
robust two- and three-dimensional (2D and 3D) self-trapped states in
Bose-Einstein condensates (BECs), which are stabilized by effective
selffirepulsion induced by quantum fluctuations around the mean-field (MF)
states [alias the Lee-Huang--Yang (LHY) effect]. The basic models are
presented, taking special care of the dimension crossover, 2D -> 3D. Recently
reported experimental results, which exhibit stable 3D and quasi-2D QDs in
binary BECs, with the inter-component attraction slightly exceeding the MF
self-repulsion in each component, and in single-component condensates of atoms
carrying permanent magnetic moments, are presented in some detail. The summary
of theoretical results is focused, chiefly, on 3D and quasi-2D QDs with
embedded vorticity, as the possibility to stabilize such states is a remarkable
prediction. Stable vortex states are presented both for QDs in free space, and
for singular but physically relevant 2D modes pulled to the center by the
inverse-square potential, with the quantum collapse suppressed by the LHY
effect.
