Directly linked to the mass of the dark matter halo where baryons form, the galaxy stellar mass (M_star) is the most important observable parameter to understand galaxy evolution. Still, we have not completely figured out how the growth of the stellar mass in galaxies happen. A fundamental problem stands at the fact that galaxies are complex systems with an intricate combination of different structural components such as bulges, discs, and bars. Each of these structures may have followed a different formation path, and therefore the separated mass growth of bulges, discs, and bars might hold the key to understand how galaxies evolved with time. In a series of papers, we have applied a new spectro-photometric decomposition code (C2D; Méndez-Abreu et al. 2019) to a sample of photometrically classified bulge+disc galaxies observed within the CALIFA IFS survey (Sánchez et al. 2016). Our approach allows us to overcome the general problem of 'structure superposition' in galaxies by separating IFS datacubes, containing the spatial and spectral information, of the individual components shaping the galaxy. The analysis of the independent bulge and disc datacubes opens a new way to analyse their spectroscopic properties. We summarise here our main results on the stellar mass growth of bulges and discs in the nearby Universe.