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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 Néel 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.