2D and 3D solitons and related states, such as quantum droplets, can appear in optical systems, atomic Bose–Einstein condensates (BECs) and liquid crystals, among other physical settings. However, multidimensional solitary states supported by the standard cubic nonlinearity tend to be strongly unstable — a property far less present in 1D systems. Thus, the central challenge is to stabilize multidimensional states, and to that end numerous approaches have been proposed over the years. Most strategies involve non-cubic nonlinearities or using various potentials, including periodic ones. Completely new directions have recently emerged in two-component BECs with spin–orbit coupling, which have been predicted to support stable 2D and metastable 3D solitons. A recent breakthrough is the creation of 3D quantum droplets. These are self-sustained states existing in two-component BECs, stabilized by the quantum fluctuations around the underlying mean-field states. Here, we review recent results in this field and outline outstanding current challenges.