The authors derive a number of new oscillation criteria for hyperbolic equations. First of all, three theorems are proved, giving sufficient conditions for oscillation of solutions of the characteristic initial value problem \[ (2.2)\quad u_{xy}+c(x,y,u)=f(x,y),\quad u_ x(x,0)=g(x),\quad u_ y(0,y)=h(y), \] in an unbounded region contained in the positive cone of \(R^ 2\). The oscillation of (2,2) is shown to be a consequence of the oscillation of certain associated ordinary differential inequalities, where oscillation is taken to mean that all solutions oscillate. The reduction to an ordinary differential equation is achieved by means of an averaging technique used by Toshida for the equation \(u_{xy}+c(x,y,u)=0.\) Several examples are given which illustrate the results. In section 3, these results are extended to hyperbolic equations in n dimensions. The equation is \((3.2)\quad u_{tt}-\Delta u+c(t,x,u)=f(t,x)\) with x in a bounded domain G, having smooth boundary \(\partial G\). The boundary condition is \(\partial u/\partial \nu =g(t,x), (t,x)\in(0,\infty)\times \partial G.\) In this case, the reduction to an ordinary differential equation is achieved by means of an averaging function. Again, sufficient conditions are obtained for oscillation of all solutions of (3.2), and several interesting examples are given. Finally, generalizations are given to noncylindrical domains, and similar oscillation theorems are obtained.