  This paper studies the situation when two 4-dimensional Lorentz manifolds
(that is, space-times) admit the same (unparametrised) geodesics, that is, when
they are projectively related. A review of some known results is given and then
the problem is considered further by treating each holonomy type in turn for
the space-time connection. It transpires that all holonomy possibilities can be
dealt with completely except the most general one and that the consequences of
two space-times being projectively related leads, in many cases, to their
associated Levi-Civita connections being identical.
