I will begin by recalling some very classical series that we usually come across in a first course in Calculus. When recast using arithmetic terminology in vogue, these series provide examples of special values of $L$-functions. An $L$-function is a function of a complex variable s attached to some interesting arithmetic or geometric data, and the special values of the $L$-function give structural information of the data to which it is attached. (A classical example to drive home this point would be Dirichlet's theorem that a certain $L$-function admitting a nonzero value when evaluated at $s=1$ implies that there are infinitely many primes in arithmetic progressions.) My aim in this talk will be to give a general idea of the governing conjectures and results in this field. Using some illustrative low-dimensional examples, I will discuss some recent results which use geometric techniques involving the cohomology of locally symmetric spaces to study the arithmetic properties of the special values of $L$-functions.