  A well known theorem of Kuratowski in 1932 states that a graph is planar if,
and only if, it does not contain a subdivision of $K_5$ or $K_{3,3}$. Wagner
proved in 1937 that if a graph other than $K_5$ does not contain any
subdivision of $K_{3,3}$ then it is planar or it admits a cut of size at most
2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph
does not contain any subdivision of $K_5$ then it is planar or it admits a cut
of size at most 4. In this paper, we give a proof of the Kelmans-Seymour
conjecture. We also discuss several related results and problems.
