Summary: In 1936 W. Burau discovered an interesting family of \(n\times n\) matrices that give a linear representation of Artin’s classical braid group \(B_n\), \(n=1,2,\ldots\). A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for \(n\leq 3\), nonfaithful for \(n\geq 5\), but the case of \(n=4\) remains open to this day, in spite of many papers on the topic. This paper introduces braid groups, describes the problem in ways that make it accessible to readers with a minimal background, reviews the literature, and makes a contribution that reinforces conjectures that the Burau representation of \(B_4\) is faithful.