Matchings were among the earliest motivations for graph theory. They subsequently remained a central goal, inspiring the development of new tools that went well beyond problems directly concerning matchings. These tools proved widely applicable, accompanying the growth of graph theory over the past century. A legendary milestone in this trajectory is W. T. Tutte, "The factorization of linear graphs," J. Lond. Math. Soc. (1), 22, no. 2, (1947), 107-111, which firmly embedded graph theory through matchings into the body of classical mathematics, in particular, linear algebra and polynomials. In this note we revisit this article presenting its original content, sketching some aspects of its impact until some recent progress, and trace one of its subsequent lines of development finally leading to a new contribution answering an open challenge and extending known results.