  Finite index subgroups of the modular group are of great arithmetic
importance. Farey symbols, introduced by Ravi Kulkarni in 1991, are a tool for
working with these groups. Given such a group $\Gamma$, a Farey symbol for
$\Gamma$ is a certain finite sequence of rational numbers (representing
vertices of a fundamental domain of $\Gamma$) together with pairing information
for the edges between the vertices. They are a compact way of encoding the
information about the group and they provide a simple way to do calculations
with the group. For example: calculating an independent set of generators and
decomposing group elements into a word in these generators, finding coset
representatives, elliptic points, and genus of the group, testing if the group
is congruence, etc. In this expository article, we will discuss Farey Symbols
and explicit algorithms for working with them.
