Deciding whether or not two curves are congruent under rotations and translations is a classical, but surprisingly subtle problem. In addition to its theoretical interest, this problem has numerous applications in computer vision and image processing, automated assembly,  signal processing, and more. To address this, as well as more general congruence problems, the signature curve parameterized by differential invariants was introduced by Calabi, Olver, Shakiban, Tannenbaum, and Haker (1998). While congruent curves have identical signatures, the converse is not true, as shown in Muso and Nicolodi (2009).  In a joint work with Eric Geiger (2021), we presented a mechanism for constructing non-congruent, non-degenerate curves with identical signatures. We also introduced a notion of the signature quiver and used it to formulate a congruence criterion for non-degenerate curves with non-simple signatures.