We consider the Cauchy problem for second order strictly hyperbolic operators when the coefficients of the principal part are not Lipschitz continuous, but only ``Log-Lipschitz'' with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis.

In particular, we study local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem.

We also give an application of the method to a continuation theorem for nonlinear wave equations where the coefficients depend on $u$: the smooth solution can be extended as long as it remains Log-Lipschitz.

Moreover we consider the case of coefficients only ``Log-Zygmund'' continuous with respect to time variable and ``Log-Lipschitz'' continuous with respect to space variables. Finally we consider the analogous problem for hyperbolic systems.