When a mathematician is faced with a sequence of numbers that one wants to understand, one typically packages them together as a generating function. For example, if one has an algebraic variety $V$ over a finite field $F_q$, a geometric object defined as the zero locus of a set of equations, one can consider the sequence of cardinalities $N_k$ over higher field extensions $F_{q^k}$. One particular generating function for the $\{N_k\}$, known as the Zeta Function of variety $V$, has lots of remarkable properties. These were conjectured by Weil in the 1940's and proven by Deligne in 1973, work which helped him earn a Fields Medal. In this talk I will give a snapshot of this work, for the case of curves, where the theory is already very rich.