Wave propagation in hyperbolic metamaterials is described by the Maxwell equations with a frequency dependent tensor of dielectric permittivity, whose eigenvalues are of different signs. In this case the problem becomes hyperbolic (Klein-Gordon equation) for a certain range of frequencies. The principal theoretical and numerical difficulty comes from the fact that this hyperbolic equation is posed in a free space, without initial conditions provided. The subject of the work is the theoretical justification of this problem. In particular, this includes the construction of a radiation condition, a well-posedness result, a limiting absorption principle and regularity estimates on the solution.