A flat surface is a surface that comes equipped with Euclidean geometry. I will begin by defining flat surfaces and giving examples. They arise in different areas of mathematics, one of which is from studying billiards in polygons. I will explain this connection and show how billiard paths give rise to what is called a linear flow on the surface; an important example of a dynamical system. I will introduce a few basic notions in the subject of dynamical systems and explain some things one can prove about these linear flows.