Liquid crystals (LCs) are matter in a state that exhibits intermediate physical properties between those of conventional liquids and those of solid crystals: they may flow like a liquid, but their molecules may be oriented along a common direction like in a crystal. The particular shape of LCs renders the molecules very sensitive to the presence of physical boundaries and to the action of magnetic or electric fields. Depending on the magnetic properties of the LC, molecules may reorient along, or normally to, the direction of the field. In particular, the sudden application of a field switches on/off the transmission of polarized light. Thus, LCs may find wide applications in computer monitors, flat-panel televisions, cell phones, calculators and watches. The classical mathematical theory of LCs (that concerns only statics) is a continuum theory based on the pioneer works by Oseen (1933), Zocher (1933) and Frank (1958). The continuum dynamical theory is instead due to the independent contributions by Ericksen (1962) and Leslie (1968). These theories use a single order parameter, called the director, a unit vector pointing along the average microscopic molecular orientation. Many phenomena in LCs fit well within the classical description. However, the transition from ordered to disordered states escapes the director theory. The classical microscopic description of defects and surface phenomena yields undesired results as well. The more recent order-tensor theory put forward by de Gennes (Nobel Prize laureate in physics in 1991) in two works (dated 1969 and 1971) focuses on the orientational probability distribution, and introduces the measures of the degree of orientation and biaxiality. This theory was reformulated rigorously by Ericksen (1991). Recent studies have shown the inadequacy of classical continuum theories to study: - the geometry and topology of active liquid crystals under confinement, - the mechanics of ultra-thin nematics deposited on curved substrates (nematic shells), - the anisotropic elasticity and dynamic relaxation of LCs. On the contrary, these phenomena can be studied by means of suitable generalizations or adaptations of existing classical models or thanks to novel more complex theories. The course aims at providing carefully crafted overviews of classical and novel continuum theories for LCs to study topological defects, equilibrium textures, active flows and acoustic wave propagation. The lectures will include surveys of relevant differential geometry and analytical methods that are essential to a proper understanding, in addition to overviews on the mathematical modelling of the subject from various perspectives. Representing a "tour d’horizon" on the physics of LCs, the presentations will highlight the efficiency of continuum theories in modelling real world phenomena. The course is addressed to doctoral students, post-doctoral researchers and academics interested in the use of continuum mechanics to model, analyse and understand the physics and mechanics of liquid crystals.