If p is a real square matrix, a Volterra multiplier is a positive diagonal matrix a such that, in the sense of quadratic forms, $ap\leqq 0$. Ever since the pioneering work of Volterra over half a century ago, it has been known that these multipliers are a significant aid in the study of stability. However, the utility of the method is diminished by the difficulty of deciding whether the multiplier exists. Here we give a number of new criteria for existence, usually under the hypothesis that p is combinatorially symmetric; that is, $p_{ij} = 0$ implies $p_{ij} = 0$. This is much weaker than the condition “$p_{ij} p_{ji} < 0$ unless $p_{ij} = p_{ji} = 0$” which has been used by Volterra and others, and greatly increases the scope of the results. Although the primary emphasis is on sufficient conditions that are easy to use, we give necessary and sufficient conditions for several cases of practical interest.