The main focus of the thesis are magnetic responses of itinerant fermions in several models relevant for the CuO2 plane in cuprate superconductors. In particular, we study the changes in magnetic spectra on passing through the pseudogap phase. First we introduce the cuprate superconductors and their rather complicated phase diagram. The pseudogap phase, appearing below a characteristic temperature T*, features a partially gapped Fermi surface and the coexistence of several fluctuating spin and charge density waves. Magnetic spectra can be divided into two groups: one shaped like the hourglass, with two branches, and the other with only one, high energy branch (in the mercury cuprate Hg1201). As suggested by experiments, this phase is most likely magnetic, so we assume it is brought on by a condensation of some magnetic mode with a plausible dispersion conforming with the upper branch of the magnetic excitations. The dynamically opened pseudogap is parametrised as a static gap at q = QAF and no explicit k-dependence is required to describe the gap shape measured in ARPES. The damping rate of this electron-magnon scattering, used here as a parameter, is required to increase with temperature. Non-interacting metallic models are shown to predict incommensurate phases, in sharp contrast with neutron scattering experiments which observe only commensurate signals at QAF at very high temperatures, close to T*. Introducing strong correlations, caused by a strong Coulomb repulsion on the copper orbitals, resolves this inconsistency and recovers the tendency towards commensurate AF order at high temperatures. A gap of any origin, opening around the antinodal parts of the Fermi surface below T* around QAF opens a spin gap in the magnetic spectrum. The part of the upper branch of magnetic excitations, just above the gap, is therefore identified as interband particle-hole excitations, rather than the usual collective modes of some ordered state. Attempts to interpret the low-temperature response as a real phase transition, by solving the Dyson equation for an AF state, result in magnetic spectra differing greatly from the experimental ones. This is consistent with the assumption that the pseudogap phase boundary is a crossover rather than real phase transition. The same constant gap function reproduces the correct wave vector for the low-temperature charge density wave order, which is higher than the antinodal nesting of the non-ineracting models. The bismuth cuprate Bi2201 may, however, feature a CDW gap opening at the antinode already at T*, therefore leaving open the question of the correct nesting vector for this class of cuprates.