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Affine vector fields in 4-dimensional Lorentz manifolds have recently been investigated in some detail by Hall and da Costa. The picture is completed in this thesis by studying the zeros of affine vector fields. Hall and da Costa show that the problem of finding affine vector fields in non-degenerately reducible 4-dimensional Lorentz manifolds can be reduced, with one exceptional case, to the problem of finding homothetic vector fields in lower dimensional manifolds. This means that the study of affine vector fields with zeros in 4-dimensional Lorentz manifolds is aided by investigating proper homothetic and Killing vector fields with zeros in 2- or 3-dimensional manifolds. To this end proper homothetic vector fields with zeros are investigated in 2- and 3-dimensional Manifolds using techniques similar to those used by Hall. It is shown that in the 2-dimensional case the zero is necessarily isolated, whereas in the 3-dimensional case the zero set may either be isolated or 1-dimensional. In the latter case the manifold is shown to be a 3-dimensional plane wave space-time, and all of the affine and conformal vector fields that it admits are found. These results are then used to determine the nature of the zero sets of affine vector fields in 4-dimensional Lorentz manifolds. The algebraic structure of the Riemann, Ricci and Weyl tensors at such zeros is also described. This work is extended by studying affine vector fields, and their zero sets, in 3-dimensional Lorentz manifolds. An investigation of the zero sets of affine vector fields in 3- and 4-dimensional positive-definite manifolds is included for comparison.