The mass and structural assembly of galaxies is still a matter of intense debate. Current theoretical models predict the existence of a linear relationship between galaxy effective radius (R_e ) and the host dark matter halo virial radius (R_h ).

By making use of accurate and transparent semi-empirical models compared to the size distributions of central galaxies from the Sloan Digital Sky Survey, I will provide robust constraints on the normalization and scatter of the Re − Rh relation in the framework of Halo Abundance Matching. Specifically, I will explore the parameter space
of models in which the relation between galaxy size and halo radius is mediated by the dynamical or structural properties of the dark matter halo. Within the same framework, I will also discuss the size evolution of Ultra Massive Galaxies, which is still poorly understood.

I will show that the data require extremely tight relations for both early-type and late type galaxies, especially for more massive galaxies. These constraints challenge models based on angular momentum conservation, which predict significantly wider distributions of galaxy sizes and no trend with stellar mass. I will also show that a constant Re-Rh relation is able to reproduce the size evolution of Ultra Massive Galaxies and provide a framework to the size evolution of galaxies with any mass.

I will conclude showing that the normalization and scatter of the size-halo radius relation of both early- and late-type galaxies might be consistent with pure merger models, though a complete picture for the structural evolution of galaxies remains elusive.