I will provide a brief introduction to translation surfaces, which are surfaces that can be constructed by gluing finitely many polygons in $R^2$ along parallel edges to form a closed surface with cone points. I will discuss the geometric notion of complete periodicity as it relates to the classification of lattice translation surfaces in genus two given by Calta and McMullen. Then I will introduce the notion of algebraic periodicity which generalizes that of complete periodicity and discuss recent results of Calta and Smillie related to algebraic periodicity.