Appears in collection : Summer School 2012 - FEUILLETAGES, COURBES PSEUDOHOLOMORPHES, APPLICATIONS

A nonsingular holomorphic foliation of codimension q on a complex manifold X is locally given by the level sets of a holomorphic submersion to the Euclidean space \mathbb C^q. If X is a Stein manifold, there also exist plenty of global foliations of this form, so long as there are no topological obstructions. More precisely, if 0< q < dim X then any q-tuple of pointwise linearly independent (1,0)-forms can be continuously deformed to a q-tuple of differentials df_1,...,df_q where f=(f_1,...,f_q) is a holomorphic submersion of X to \mathbb C^q. Such a submersion f always exists if q is no more than the integer part of 2n/3. More generally, if E is a complex vector subbundle of the tangent bundle TX such that TX\backslash E is a flat bundle, then E is homotopic (through complex vector subbundles of TX) to an integrable subbundle, i.e., to the tangent bundle of a nonsingular holomorphic foliation on X. I will prove these results and discuss open problems, the most interesting one of them being related to a conjecture of Bogomolov.