Theoretical results of studying thermodynamic and structural characteristics of free droplets and bubbles around nanosized solid heterogeneous spherical inclusions using phenomenological thermodynamic approach and various versions of the molecular density functional method are reviewed. In the case of a droplet in an undersaturated or supersaturated vapor, the central solid particle is assumed to be lyophilic and can be either electrically charged, or uncharged. In the case of a vapor bubble in a stretched liquid, the central particle is assumed to be lyophobic and, as in the previous case, can be electrically charged or uncharged. The structure of droplets and bubbles is described by equilibrium molecular density profiles. Thermodynamic characteristics imply the chemical potential of molecules in a droplet or bubble as a function of their size, the work of formation of equilibrium droplets or bubbles as a function of the chemical potential of molecules in the system, and the surface tension and disjoining pressure in droplets or bubbles as functions of its radius.