A peculiar module is a certain algebraic invariant of 4-ended tangles that I developed in my PhD thesis as a tool for studying the local behaviour of Heegaard Floer homology for knots and links. I will briefly explain its construction and describe its classification in terms of immersed curves on a 4-punctured sphere as well as a glueing formula. Finally, I will discuss some applications, such as rational tangle detection, skein relations and mutation symmetries.