The immersed Finite Element Method (FEM), also known as the unfitted or cut FEM, and the Material Point Method (MPM) are two classes of methods that can handle complex and evolving geometries by decoupling the domain description from the mesh discretisation. However, this introduces a number of issues that are not faced by conventional FEMs, where the geometry is explicitly linked to the nodes of the finite element mesh. The immersed FEM and the MPM have developed in relative isolation but there are several parallels in terms of boundary representation, imposition of boundary conditions, integration and conditioning of the resulting system of equations. There are also clear differences. This contribution explores the similarities and differences between the methods, focusing on areas where the two communities could capitalise on knowledge transfer for mutual benefit.