This thesis is divided into three part, the first part concerns metric spaces and specically length spaces where the existence of shortest path between points is the main focus. In the second part, an example of a length space, the Riemannian geometry will be given. Here both a classical approach to Riemannian geometry will be given together with specic results when considered as a metric space. In the third part, the Finsler geometry will be examined both with a classical approach and trying to deal with it as a metric space.