Strongly nonlinear waves (SNWs) and extreme states of matter are key physical concepts. A SNW is a wave whose amplitude is on the order of the material's internal strength limit. When a shock front appears at a distance of 10∧2-10∧3 wavelengths in water, nonlinearity is weak but strongly expressed. The acoustic pressure is 10∧5-10∧6 Pa, much less than the internal pressure 2.2*10∧9 Pa. However, impurities decrease the breaking strength, and waves creates bubbles at smaller pressures. An explosive wave is also a SNW, breaking solids. For weakly nonlinear waves (WNWs) the equation of state can be expanded in power or functional series. However, these cannot be used in 3 cases. First, if the equation contains singularities, like for "clapping" and Hertz nonlinearities of heterogeneous solids. Secondly, if the series is divergent. Thirdly, when the linear term is absent and the higher nonlinearities dominate. Such SNWs appear in mechanics and in quantum field theory.