  We consider inverse problems in space-time $(M, g)$, a $4$-dimensional
Lorentzian manifold. For semilinear wave equations $\square_g u + H(x, u) = f$,
where $\square_g$ denotes the usual Laplace-Beltrami operator, we prove that
the source-to-solution map $L: f \rightarrow u|_V$, where $V$ is a neighborhood
of a time-like geodesic $\mu$, determines the topological, differentiable
structure and the conformal class of the metric of the space-time in the
maximal set where waves can propagate from $\mu$ and return back. Moreover, on
a given space-time $(M, g)$, the source-to-solution map determines some
coefficients of the Taylor expansion of $H$ in $u$.
