  This PhD thesis studies the broken ray transform, a generalization of the
geodesic X-ray transform where geodesics are replaced with broken rays that
reflect on a part of the boundary. The fundamental question is whether this
transform is injective. We employ four different methods to approach this
question, and each of them gives interesting results.
  Direct calculation can be used in a ball, where the geometry is particularly
simple. If the reflecting part of the boundary is (piecewise) flat, a
reflection argument can be used to reduce the problem to the usual X-ray
transform. In some geometries one can use broken rays near the boundary to
determine the values of the unknown function at the reflector, and even
construct its Taylor series. One can also use energy estimates -- which in this
context are known as Pestov identities -- to show injectivity in the presence
of one convex reflecting obstacle. Many of these methods work also on
Riemannian manifolds.
  We also discuss the periodic broken ray transform, where the integrals are
taken over periodic broken rays. The broken ray transform and its periodic
version have applications in other inverse problems, including Calder\'on's
problem and problems related to spectral geometry.
  A more detailed abstract can be found in the PDF file. This version only
contains the introductory parts of the thesis. The full thesis also contains
five articles.
