In the field \(\mathbb{C}((t))\), a metric can be given based on the order of vanishing. However this space is totally disconnected. I will begin with the construction of the Berkovich projective line over a non-Archimedean field closely related with \(\mathbb{C}((t))\). The Berkovich projective line is connected and Hausdorff, and rational maps can be extended naturally there. I will give an example-based introduction to those rational maps. Time permitting, I will talk briefly about dynamics.