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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(Γ(𝑅)). It is the (undirected) graph with all elements of 𝑅 as vertices and, for distinct 𝑥, 𝑦 ∈ 𝑅, the vertices 𝑥 and 𝑦 are adjacent if and only if 𝑥 + 𝑦 ∈ 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.