Random Maps and Their Scaling Limits

We review some aspects of scaling limits of random planar maps; which can be considered as a model of a continuous random surface, and have driven much interest in the recent years. As a start, we will treat in a relatively detailed fashion the well-known convergence of uniform plane trees to the Brownian Continuum Random Tree. We will put a special emphasis on the fractal properties of the random metric spaces that are involved; by giving a detailed proof of the calculation of the Hausdorff dimension of the scaling limits.