Significant advances in contact mechanics have been achieved since the first theoretical derivations a few centuries ago, primarily associated with contact problems in statics and dynamics involving friction, adhesion, wear, roughness, heat or electric conduction, and also with materials not only linear elastic. Principles of contacts mechanics can be applied in many traditional mechanical engineering areas such as locomotive wheel-rail contact, coupling devices, braking systems, tires, bearings, combustion engines, mechanical linkages, gasket seals, metal forming, ultrasonic welding, electrical contacts, and many others. Current challenges in the field regard the extension of contact mechanics methodologies to the micro- and the nano-scales, to coupled multi-field problems, and the application to finite elasticity. The main objective of this course is to convey, in a self-contained manner, the fundamental concepts for the classification of the types of contact, the mathematical methods for the formulation of the contact problems, and the numerical procedures required for their solution. In addition to the methodologies, a wide class of applications will be covered, including contact problems in mechanical engineering, microelectronics and nanomechanics. A taxonomy of contacts and the half-space solutions for linear elastic problems will be provided. For the class of complete contacts, asymptotic methods are formulated and applied to mechanical engineering problems. Further concepts for modelling contact problems with friction and partial slip will also be provided. Methods to formulate contact problems in the presence of coupled fields, such as thermo-elastic and electro-elastic contact problems, are then presented together with applications to mechanical engineering and microelectronics. An overview on numerical methods for the approximate solution of contact problems will also be provided (boundary element method, finite element method, molecular dynamics) with attention to complex problems characterized by multi-scale roughness, emphasizing the advantages and disadvantages of each technique. Finally, advanced contact problems involving lubrication, roughness and interaction between bodies undergoing finite deformation will complete the course. Each set of lectures will be designed to convey a strong background on theory and numerical methods, with also in-depth treatment of cutting-edge research topics and applications. The final aim of this intensive course is to provide a compact yet comprehensive overview of contact mechanics in technology and its current challenges. The lectures are primarily tailored for doctoral students of applied mathematics, mechanics, engineering and physics with a strong research interest in theoretical modeling, numerical simulation and experimental characterization of contact problems in technology. They are also suited for young and senior researchers in the above-mentioned and neighboring fields working in academia or in private research and development centers, interested in gaining a compact yet comprehensive overview of contact mechanics from its fundamental mathematical background, to the computational methods and the experimental techniques available for the solution of contact problems.