It is addressed to beginners as well as specialists in these fields. The contributions are to a large extent self-contained. Key topics include: - low regularity solutions to the local Cauchy problem associated with wave maps; local well-posedness, non-uniqueness and ill-posedness results are proved - coupled systems of wave equations with different speeds of propagation; here pointwise decay estimates for solutions in spaces with hyperbolic weights come in - damped wave equations in exterior domains; the energy method is combined with the geometry of the exterior domain; for the critical part of the boundary a restricted localized effective dissipation is employed - the phenomenon of parametric resonance for wave map type equations; the influence of time-dependent oscillations on the existence of global small data solutions is studied - a unified approach to attack degenerate hyperbolic problems as weakly hyperbolic ones and Cauchy problems for strictly hyperbolic equations with non-Lipschitz coefficients - weakly hyperbolic Cauchy problems with finite time degeneracy; the precise loss of regularity depending on the spatial variables is determined; the main step is to find the correct class of pseudodifferential symbols and to establish a calculus which contains a symmetrizer.