We review recent activity on how geometrically confining polymers in colloidal nanoparticles interfere with glassy dynamics and crystallization kinetics. In doing so, we emphasize the specific features of colloidal polymer nanoparticles. In these systems the presence of a large interfacial area, an aspect shared with other types of confinement, constitutes a factor of utmost importance in determining deviations in glassy dynamics and crystallization kinetics from bulk behavior. Furthermore, the relevance of the curvature at the polymer surface, which is a specific feature of colloidal polymer nanoparticles, is discussed. The suppression of degrees of freedom is stigmatized as responsible for the increase in the glass transition temperature observed in some specific cases of colloidal polymer nanoparticles and the slowing down of crystallization.