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.