  This thesis deals with the Hubbard model as prototypical model to describe
the physics of electrons in the two-dimensional copper-oxide planes of
high-$T_c$ cuprates. To get approximate solutions, we employ functional
renormalization group (fRG) and dynamical mean-field theory (DMFT) methods.
  We deal with the problem of identifying bosonic fluctuations in the vertex
function, exhibiting an intricate dependence on momenta and frequencies already
at moderate coupling. In the normal, paramagnetic phase, the goal is achieved
by employing the recently introduced single-boson exchange decomposition. In
the symmetry-broken phases, we reformulate the previously introduced
combination of fRG with mean-field theory by explicitly introducing a bosonic
field.
  A widely discussed and challenging problem is the emergence of a pseudogap in
the Hubbard model. In this thesis we assume this phase to be characterized by
strong magnetic fluctuations. Following previous works, we fractionalize the
electron in a chargon, carrying the electron's charge, and a charge neutral
spinon, which represents local fluctuations of the spin orientation. The
chargons undergo N\'eel or spiral magnetic order below a density-dependent
transition temperature $T^*$. We compute DC charge transport coefficients for
the chargons, and obtain a pronounced drop in the charge carrier density across
the magnetic-to-paramagnetic transition. Directional fluctuations of the spin
orientation are described by a non-linear sigma model and prevent long-range
ordering of the electrons at any finite temperature. The phase where the
chargon degrees of freedom are magnetically ordered shares many features with
the pseudogap regime observed in high-$T_c$ cuprates: a strong reduction of the
charge carrier density, a spin gap, Fermi arcs, and electronic nematicity.
