We use scaling theory to derive the time dependence of the mean-square displacement of a probe nanoparticle of size d experiencing thermal motion in polymer solutions and melts. Particles with size smaller than solution correlation length xi undergo ordinary diffusion ( similar to t) with diffusion coefficient similar to that in pure solvent. The motion of particles of intermediate size (xi < d < a), where a is the tube diameter for entangled polymer liquids, is subdiffusive ( similar to t(1/2)) at short time scales since their motion is affected by subsections of polymer chains. At long time scales the motion of these particles is diffusive, and their diffusion coefficient is determined by the effective viscosity of a polymer liquid with chains of size comparable to the particle diameter d. The motion of particles larger than the tube diameter a at time scales shorter than the relaxation time tau(e) of an entanglement strand is similar to the motion of particles of intermediate size. At longer time scales (t > tau(e)) large particles (d > a) are trapped by entanglement mesh, and to move further they have to wait for the surrounding polymer chains to relax at the reptation time scale tau(rep). At longer times t > tau(rep), the motion of such large particles (d > a) is diffusive with diffusion coefficient determined by the bulk viscosity of the entangled polymer liquids. Our predictions are in agreement with the results of experiments and computer simulations.