Solid, liquid, and gaseous states of matter can exist and acquire unique properties when reduced in size into a nanometer domain. This Colloquium explores the approaches to produce plasmas with nanometer dimensions and the arising physical phenomena and properties associated with this extreme, nonequilibrium state of matter. Analysis of the spatial confinement, coupling, ideality, and degeneracy criteria lead to the possibilities to produce transient nanoplasma states near, in, and from solids by using ultrafast photoexcitation. These states arise through the interplay of nonequilibrium, many-body Coulomb interactions, thermal, and nonthermal effects. Examples include photoexcited electron-hole plasmas in semiconductors, transient solid-to-plasma states including warm dense matter, nanoplasmas produced by interaction of nanoclusters and nanoparticles with intense radiation, nanoplasmas in high-energy ion tracks within solids, nanoplasmas in relativistic regime, and others. Physical phenomena arising due to the localization of high-energy densities to microscales and nanoscales and their potential applications are discussed.