The author studies the Goursat problem for semilinear wave equations in multidimensional space \[ \square u= f(x, t, u),\;(x, t)\in \Gamma,\;u|_{\partial\Gamma}= 0, \] where \((x, t)\in \mathbb{R}^{n+ 1}\), \(n\geq 2\), \(f\) is a \(C^\infty\) function of its arguments, and \(\Gamma= \{(x, t)\mid |x|\leq t\}\), \(\partial\Gamma= \{(x,t)\mid|x|= t\}\). Using the method of ``vector fields'' and techniques of microlocalization, the author proves that the solution of the Goursat problem is Lipschitz and is smooth away from the boundary of \(\Gamma\).