Summary: A set \(M\) of edges of a graph \(G\) is a matching if no two edges in \(M\) are incident to the same vertex. The matching number of \(G\) is the maximum cardinality of a matching of \(G\). A set \(S\) of vertices in \(G\) is a total dominating set of \(G\) if every vertex of \(G\) is adjacent to some vertex in \(S\). The minimum cardinality of a total dominating set of \(G\) is the total domination number of \(G\). If \(G\) does not contain \(K_{1,3}\) as an induced subgraph, then \(G\) is said to be claw-free. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number. In this paper, we use transversals in hypergraphs to characterize connected claw-free graphs with minimum degree at least three that have equal total domination and matching numbers.