We review the state of the art and recently obtained theoretical and experimental results for two-and three-dimensional (2D and 3D) solitons and related states, such as quantum droplets, in optical systems, atomic Bose-Einstein condensates (BECs), and other fields -in particular, liquid crystals. The central challenge is avoiding the trend of 2D and 3D solitary states supported by the ubiquitous cubic nonlinearity to be strongly unstable -a property far less present in one-dimensional systems. Many possibilities for the stabilization of multi-dimensional states have been theoretically proposed over the years. Most strategies involve non-cubic nonlinearities or using different sorts of potentials, including periodic ones. New important avenues have arisen recently in systems based on two-component BEC with spin-orbit coupling, which have been predicted to support stable 2D and metastable 3D solitons. An important recent breakthrough is the creation of 3D quantum droplets. These are self-sustained states existing in two-component BECs, which are stabilized by the presence of Lee-Hung-Yang quantum fluctuations around the underlying mean-field states. By and large, multi-dimensional geometries afford unique opportunities to explore the existence of complex self-sustained states, including topologically rich ones, which by their very nature are not possible in one-dimensional geometries. In addition to vortex solitons, these are hopfions, skyrmions, and hybrid vortex-antivortex complexes, which have been predicted in different models. Here we review recent landmark findings in this field and outline outstanding open challenges. List of acronyms:1D one-dimensional 2D two dimensional 3D three dimensional BEC Bose-Einstein condensate FF fundamental frequency GPE Gross-Pitaevskii equation GVD group-velocity dispersion LHY Lee-Huang-Yang (correction to the mean-field dynamics of BEC) MF mean field MM mixed mode NLSE nonlinear Schrödinger equation SH second harmonic SOC spin-orbit coupling SV semi-vortex TS Townes' soliton VK Vakhitov-Kolokolov (stability criterion)

No abstract

We investigate the existence and stability of three-dimensional spatiotemporal solitons in self-focusing cubic Kerr-type optical media with an imprinted two-dimensional harmonic transverse modulation of the refractive index. We demonstrate that two-dimensional photonic Kerr-type nonlinear lattices can support stable one-parameter families of three-dimensional spatiotemporal solitons provided that their energy is within a certain interval and the strength of the lattice potential, which is proportional to the refractive index modulation depth, is above a certain threshold value.