A revolution is currently taking place in physics and engineering through the manufacture of metamaterials and metasurfaces with the aim of achieving full control of waves. This was made possible by conceiving and designing new materials whose macroscopic behavior results from a specific structure, often periodic, at the microscopic scale. Typical examples are the band-gap materials and the double negativity metamaterials which are based on the local resonance (of the Mie, Minnaert or Helmholtz’s type) of a subwavelength building block repeated periodically. These materials have found numerous practical applications among which cloaking, lensing, super-resolution, quantitative imaging in the near field, shielding, perfect absorption. In acoustics, a well known example is inspired by the pots used in the ancient greek theaters and later on in churches and mosques to control the acoustic of the places; these pots are Helmholt’z resonators and they are thought nowadays as the key to soundproof wall design. More recent examples are related to the shielding of regions from waves tanks to  recent advances  in the design of metamaterial based devices. This is the case of the so-called anti-seismic wedge formed by a forest of trees able to convert the destructive surface waves into mainly harmless downward propagating bulk waves. Another striking example is the design of gigantic « wave breakers » to surround and protect a region from swell and stormy waves. The efficiency of such belts becomes a vital prerequisite to viable and sustainable floating cities which are sought in the foreseeable future.   The purpose of this course is to provide an introduction to well known techniques and to introduce more advanced, state-of-the-art techniques, able to tackle the challenges of the metamaterials by providing a mathematical framework able to explain the observed extraordinary properties of meta-structures and useful to help optimize these properties. These include transformation optics/ transformation elastodynamics, classical and modern homo-genization methodologies, asymptotic and spectral analysis, variational methods, layer potential techniques, as well as modern multi-mathematics. Particular attention will be paid to present practical applications that illustrate the workings and effectiveness of the introduced techniques and discussions on the breakthroughs and the remaining open questions on topical issues in acoustics, electromagnetism, elasticity and in the context of water waves. Bearing these objectives in mind, academic experimental studies as well as commercial devices will be presented during the course. Overall this is a joint effort from a diverse group of lecturers working on different aspects of metamaterial modelling, to report the current state-of-the-art in the field and form a collaborative network and shared knowledge platform in an area where there is still considerable room for research. The course is addressed to a broad public: graduate students, doctoral students, young researchers and practicing engineers. Since a major part of the course will be devoted to theoretical and numerical modelling, a sound mathematical basis is expected and basic knowledge of wave theory is recommended. The topics explore the applications in Engineering and Physics, showing the interconnections with acoustics, electromagnetism, elasticity, water waves that are normally treated as independent topics. Both theoreticians and experimentalists from the academic and industrial sectors are expected to gain useful knowledge from attending the course.