  This concise review aims to provide a summary of the most relevant recent
experimental and theoretical results for solitons, i.e., self-trapped bound
states of nonlinear waves, in two- and three-dimensional (2D and 3D) media. In
comparison with commonly known one-dimensional solitons, which are, normally,
stable modes, a challenging problem is the propensity of 2D and 3D solitons to
instability, caused by the occurrence of the critical or supercritical wave
collapse (catastrophic self-compression) in the same spatial dimension. A
remarkable feature of multidimensional solitons is their ability to carry
vorticity; however, 2D vortex rings and 3D vortex tori are subject to strong
splitting instability. Therefore, it is natural to categorize the basic results
according to physically relevant settings which make it possible to maintain
stability of fundamental (non-topological) and vortex solitons against the
collapse and splitting, respectively. The present review is focused on schemes
that were recently elaborated in terms of Bose-Einstein condensates and similar
photonic setups. These are two-component systems with spin-orbit coupling, and
ones stabilized by the beyond-mean-field Lee-Huang-Yang effect. The latter
setting has been implemented experimentally, giving rise to stable self-trapped
quasi-2D and 3D "quantum droplets".
