A key object of study in geometric group theory is the mapping class group (MCG) of a given surface (say closed and orientable for simplicity), defined as the group of self-homeomorphisms (assume orientation-preserving) modulo isotopy. In the case of a torus T, the classification comes without too much hassle from considering linear algebra on the universal cover: each element of MCG(T) is periodic, reducible (preserves an essential simple closed curve), or Anosov (preserves a foliation). For a surface S of genus g > 1, the situation is more complicated, and Jakob Nielsen treated this case using geodesic laminations (generalizations of closed geodesics on S) in the early 20th century. William Thurston shed further light on this using independent machinery (singular foliations) in the 1970´s, and we can now think of the classification as follows: each element of MCG(S) is periodic, reducible (preserves a finite collection of simple closed geodesics), or pseudo-Anosov (preserves a geodesic lamination which is extremal in a precise sense). I will present a combinatorially flavored proof of this classification, further details of which can be found in, for example, Danny Calegari´s "Foliations and the geometry of 3-manifolds" and Handel-Thurston´s "New proofs of some results of Nielsen".