Harmonic maps between Riemannian manifolds were first established by James Eells and Joseph H. Sampson 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. Biharmonic maps, which generalize harmonic maps, were introduced by Guoying Jiang in 1986. The book presents an overview of the important developments made in these fields since their inception. Furthermore, it introduces biwave maps, which generalize wave maps and were first studied by the author in 2009, and bi-Yang-Mills fields, which generalize Yang-Mills fields and were first investigated by Toshiyuki Ichiyama, Jun-Ichi Inoguchi, and Hajime Urakawa in 2008. Other topics discussed include exponential harmonic maps, exponential wave maps, and exponential Yang-Mills fields.