A graph G is said to be (p1,p2) be-embeddable if there exist two subgraphs H1 and H2 of G with H1∪H2=G so that H1 is embedded on Sp1 and H2 is embedded on Sp2 where Sp1 and Sp2 are orientable surfaces of genera p1 and p2 respectively. Several infinite families of bi-embeddings of complete graphs are exhibited. Results from a computer search for additional bi-embeddings are also included. Finally, an unsolved problem is presented.