  Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this
paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the
(undirected) graph with all elements of $R$ as vertices, and for distinct $x,
y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y\in\Z(R)$.
We investigate properties of the total graph of $R$ and determine all
isomorphism classes of finite commutative rings whose total graph has genus at
most one (i.e., a planar or toroidal graph). In addition, it is shown that,
given a positive integer $g$, there are only finitely many finite rings whose
total graph has genus $g$.
