This review summarizes recent theoretical and experimental findings on quantum droplets (QDs), robust self-trapped states in Bose-Einstein condensates (BECs) stabilized by the Lee-Huang-Yang (LHY) effect. QDs can exist in two- and three-dimensional (2D and 3D) forms, stabilized by effective self-repulsion induced by quantum fluctuations around mean-field (MF) states. The basic models, including the LHY correction to the MF dynamics, are presented, with a focus on the dimensional crossover from 2D to 3D. Experimental results show stable 3D and quasi-2D QDs in binary BECs with inter-component attraction slightly exceeding the MF self-repulsion, and in single-component condensates of atoms with permanent magnetic moments. Theoretical results highlight the stability of 3D and quasi-2D QDs with embedded vorticity, including stable vortex states in free space and in singular 2D modes pulled to the center by an inverse-square potential, where quantum collapse is suppressed by the LHY effect. The review concludes with a discussion of related topics and future directions.This review summarizes recent theoretical and experimental findings on quantum droplets (QDs), robust self-trapped states in Bose-Einstein condensates (BECs) stabilized by the Lee-Huang-Yang (LHY) effect. QDs can exist in two- and three-dimensional (2D and 3D) forms, stabilized by effective self-repulsion induced by quantum fluctuations around mean-field (MF) states. The basic models, including the LHY correction to the MF dynamics, are presented, with a focus on the dimensional crossover from 2D to 3D. Experimental results show stable 3D and quasi-2D QDs in binary BECs with inter-component attraction slightly exceeding the MF self-repulsion, and in single-component condensates of atoms with permanent magnetic moments. Theoretical results highlight the stability of 3D and quasi-2D QDs with embedded vorticity, including stable vortex states in free space and in singular 2D modes pulled to the center by an inverse-square potential, where quantum collapse is suppressed by the LHY effect. The review concludes with a discussion of related topics and future directions.