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The objective of this thesis is to further the understanding of the time inversion property for self-similar Markov processes. In particular, we focus upon seeking a full characterisation of the class of processes that enjoy the time inversion property.\ud \ud The first chapter in this thesis is a review of current literature in the areas that we use in the sequel. Chapter 2 provides a full characterisation of processes enjoying the time inversion property on R up to certain restrictions. Namely, we show that on R+, the only processes that enjoy the time inversion property are Bessel processes in the wide sense. Extending this characterisation to R, we show that we are necessarily restricted to variations of Bessel and Dunkl processes. We then give an expression of the semigroup density that all processes with the time inversion property must satisfy. In Chapter 3, we extend some of these results to Rn. We provide a restriction on the jump measure of processes with the time inversion property and show that ^ρ(Rt) is necessarily a Bessel process for a process Rt with the time inversion property and a defined function ^ρ. Finally, Chapter 4 extends the work of Vuolle-Apiala [2012] on the skew product representation and presents a methodology by which one can construct examples of processes with the time inversion property. This leads to several examples of particular interest.