Given a periodic graph, a pair of periodic functions on the vertices and edges gives rise to a discrete periodic operator. Studying the spectrum of these operators is one of the most classical directions in mathematical physics. Floquet theory reveals that many properties of the spectrum can be studied through the Bloch variety, an algebraic variety, thereby inviting the application of tools from algebraic geometry. In this talk, we will introduce these objects and discuss the recent surge in the use of algebraic methods in the spectral theory of these operators.