It is shown that the wave equation &psgr;xx−&psgr;yy=0 for the field &psgr;(x,y) in the domainR(xy) can be transformed into a wave equation &PSgr;&xgr;&xgr;−&PSgr;&eegr;&eegr;=0 for the field &PSgr;(&xgr;,&eegr;) in the domainS(&xgr;&eegr;). The transformation is accomplished through a complex functionF(x,y)=&xgr;(x,y) +i&eegr;(x,y), which is not analytic. For the transformation to exist, the real transformation functions &xgr;=&xgr;(x,y) and &eegr;=&eegr;(x,y) have to satisfy wave equations in the domainR(xy) and the first‐order partial equations &xgr;x=±&eegr;yand &xgr;y=±&eegr;x[‘‘±’’ distinguishes transformations of the first (+) and second (−) kinds]. Thus, the hyperbolic transformation theory is different from the conformal mapping theory, where the real transformation functions satisfy the Laplace equation and the Cauchy–Riemann conditions. As applications, the linear Lorentz transformation and nonlinear mappings of time‐varying regions into fixed domains are discussed as solutions of the indicated partial differential equations. Furthermore, an initial‐boundary‐value problem for the electromagnetic wave equation with moving boundary condition is solved analytically (compression of microwaves in an imploding resonator cavity).