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Harmonic maps between Riemannian manifolds were first established in 1964. Wave maps are harmonic maps on Minkowski spaces and have been studied since the 1990s. Yang-Mills fields, the critical points of Yang-Mills functionals of connections whose curvature tensors are harmonic, were explored by a few physicists in the 1950s, and biharmonic maps (generalizing harmonic maps) were introduced in 1986. The book presents an overview of the important developments made in these fields since they first came up. Furthermore, it introduces biwave maps (generalizing wave maps) which were first studied in 2009, and bi-Yang-Mills fields (generalizing Yang-Mills fields) first investigated in 2008. Other topics discussed are exponential harmonic maps, exponential wave maps and exponential Yang-Mills fields.