  We prove that the topology, smooth structure, and metric of a compact
Lorentzian manifold with boundary is uniquely determined by data at the
boundary. The data consists of the lengths and directions of future-directed
once-broken geodesics connecting points on the boundary, which are first
timelike and then lightlike. This requires the strong causality condition and a
weak convexity assumption, but it holds without any assumptions about conjugate
points. With an additional convexity assumption we prove the analogous
statement for future-directed once-broken timelike geodesics.
  If there are no conjugate points and lightlike geodesics never refocus, the
analogous data using lightlike geodesics and once-broken lightlike geodesics
may be used to reconstruct the manifold up to a conformal factor. This is a
corollary of a result which shows that the conformal class is determined by the
collection of sets of future-directed lightlike vectors at the boundary which
give geodesics which all intersect in a single point.
