In this talk, we will discuss the dynamics on Teichmüller space and moduli space of square-tiled surfaces. For square-tiled surfaces, one can explicitly write down the $SL(2,\mathbb{R})$-orbit on the moduli space. To study the dynamics of Teichmüller flow of the $SL(2,\mathbb{R})$-action, we study its derivative, namely the Kontsevich--Zorich cocycle (KZ cocycle). In this talk, we will define what a KZ cocycle is, and by following explicit examples, we will show how one can compute the KZ monodromy. This is part of an ongoing work with Anthony Sanchez.