The Teichmuller space with the Thurston metric and Outer Space with the Lipschitz metric are two examples of spaces with an asymmetric metric i.e. d(x,y)
eq d(y,x). The latter case is also incomplete: There exist Cauchy sequences that do not have a limit. We develop the theory of the completion of an asymmetric space and give lots of examples. Time permitting we will describe the case of Outer Space.