  We study the cosmological build-up of pseudobulges using the LGalaxies
semi-analytical model for galaxy formation with a new approach for following
separately the assembly of classical bulges and pseudobulges. Classical bulges
are assumed to be the result of violent processes (i.e., mergers and
starbursts), while the formation of pseudobulges is connected to the secular
growth of disks. We apply the model to both the Millennium and the Millennium
II simulations, in order to study our results across a wide range of stellar
masses ($10^{7} - 10^{11.5} M_{\odot}$). We find that $z=0$ pseudobulges mainly
reside in galaxies of $ \rm M_{stellar} \sim 10^{10} - 10^{10.5} M_{\odot}$
($\rm M_{halo} \sim 10^{11.5}-10^{12} M_{\odot}$) and we recover structural
properties of these objects (e.g., sizes and bulge-to-total ratios) that are in
good agreement with observational results. Tracing their formation history, we
find that pseudobulges assembled in galaxies with a very quiet merger history,
as opposed to the host galaxies of classical bulges. Regarding the bulge
structure, we find that $\sim$ 30\% of the galaxies with a predominant
pseudobulge feature a composite structure, hosting both a pseudo and a
classical bulge component. The classical component typically constitutes
$\sim$10\% of the total bulge galaxy mass. When looking at the properties of
the host galaxies, we find that $z = 0$ pseudobulges are hosted by main
sequence galaxies, characterized by a stellar population which is generally
younger compared to the one of the hosts of classical bulges.
