  In this paper, we consider a number of results and seven conjectures on
properly edge-coloured (PC) paths and cycles in edge-coloured multigraphs. We
overview some known results and prove new ones. In particular, we consider a
family of transformations of an edge-coloured multigraph $G$ into an ordinary
graph that allow us to check the existence PC cycles and PC $(s,t)$-paths in
$G$ and, if they exist, to find shortest ones among them. We raise a problem of
finding the optimal transformation and consider a possible solution to the
problem.
