Abstract: This is a report on the initial stages of some work that I am doing with Prof. P.G. LeFloch (Paris VI, CNRS). Our aim is to develop techniques for estimating the injectivity radius of Lorentzian manifolds similar in nature to techniques due to Cheeger, Gromov and Taylor for Riemannian manifolds. In this talk, I will tell you about the relevant Riemannian results (e.g. the Rauch comparison theorem, Toponogov comparison theorem, volume comparison and Cheeger-Gromov-Taylor results) and to what extent we can (currently) find corresponding results for Lorentzian manifolds.