Let X1, X2 denote positive heavy-tailed random variables with continuous marginal distribution functions F1 and F2, respectively. The asymptotic behavior of the tail of X1+X2 is studied in a general copula framework and some bounds and extremal properties are provided. For more specific assumptions on F1, F2 and the underlying dependence structure of X1 and X2, we survey explicit asymptotic results available in the literature and add several new cases.