In this document, I present a personal view on the heavy-fermion problem, within a phenomenological approach guided by experiments. This review presents a set of historical works which established the ground bases of the thematic during the last decades. An exhaustive and systematic approach is privileged. After a general presentation in Chapter 2, the properties of heavy-fermion paramagnets, antiferromagnets, and ferromagnets are considered in Chapters 3, 4, and 5, respectively. Chapters 6 and 7 are dedicated to two specific compounds, URu$_2$Si$_2$ for which a hidden-order phase constitutes a more-than-thirty-years-old unsolved mystery, and UTe$_2$, where multiple superconducting phases have been discovered in the last two years. Experiments performed using a panel of techniques ranging from microscopic (neutron scattering, NMR, etc.) to thermodynamic (specific heat, magnetization, etc.) and transport (electrical resistivity, etc.) probes, under extreme conditions of low temperatures, intense magnetic fields and high pressures, are reviewed. They show that magnetism plays a central role in the quantum critical properties of heavy-fermion systems. An emphasis is given to the intersite magnetic fluctuations, presented as the driving force for a heavy Fermi liquid, precursor of quantum magnetic criticality ending in magnetically-ordered phases. They are also suspected to drive an unconventional mechanism for superconductivity, which develops in the vicinity of quantum magnetic phase transitions induced under pressure or magnetic field. The appearance of magnetic fluctuations and ultimately magnetic order in heavy-fermion compounds occurs in a nearly-integer-valence regime, in which $f$ electrons have a dual itinerant-localized character. Fermi-surface and valence studies, which give complementary information about this duality, are also considered.
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