The Jordan-Hölder theorem for groups goes back to 1870.
However, the real reason of this theorem lies in the
theory of planar semimodular lattices. This discovery
lead G. Gratzer and J.B. Nation [2010] to a strengthening of
the Jordan-Hölder theorem, which was strengthened further by
G. Czédli and E.T. Schmidt [2011]. These results together
with earlier papers by G. Grätzer and E. Knapp initiated
an intensive study of planar semimodular lattices; finiteness
is always assumed.
  A particular interest is paid to the slim
ones; a lattice is slim if the order (= poset) of its
join-irreducible elements contains no three-element antichain.
Firstly because the decsription of all planar semimodular
lattices reduces to that of the slim ones. Secondly,
because it is the slim semimodular lattices that are
associated with two composition series of groups. Thirdly,
because they are the ones we can really describe.
  In the talk, several structural descriptions of slim
semimodular lattices, due to G. Gratzer, E. Knapp,
E.T. Schmidt and G. Czédli, will be presented. These
descriptions led G. Czédli, T. Dékány, L. Ozsvárt, N. Szakács,
and B. Udvari to enumerative combinatorial results on these
lattices.
  This talk could be an appetizer to the "Planar Semimodular
Lattices and Their Diagrams" chapter, which we are writing
with G. Gratzer to the "Lattice Theory: Special Topics and
Applications" book (planned to appear in 2013).