AbstractAn embedding of graph G into graph H is by defenition an isomorphism of G onto a subgraph of H. It is shown in this paper that every unicycle U embeds in its line-graph L(U), and that every other connected graph that embeds in its own line-graph may be constructed from such an embedded unicycle in a natural way. A complete, explicit classification is also given of the graphs G that admit a special kind of embedding into L(G), called an incidence embedding. Moreover, the question of which graphs G are isomorphic to induced subgraphs G' of their line-graphs is shown to be formally equivalent to the original, more general question, where G' is not required to be induced.