Foliations have played a key role in several important developments in higher dimensional algebraic geometry. However, systematic efforts to understand their birational geometry have begun more recently. The study of birational geometry of foliated surfaces was started by Brunella and McQuillan. It was also observed that some aspects of the foliated minimal model program (MMP) are quite different from the classical one. Building on the pioneering work of Cano who proved the existence of foliated log resolution for corank one foliations in dimension three, the MMP for such foliations with mild (non dicritical) singularities was established in a series of papers by Cascini, Spicer and Svaldi. When the foliation has strictly log canonical (dicritical) singularities, one requires to use other techniques. Considering such foliations becomes essential for studying the birational geometry of Fano foliations and moduli theory of corank one foliations of general type. I report on a joint work with Roktim Mascharak where we have established the full log canonical MMP for corank one foliations. I'll try to briefly indicate how these results can be extended to the setting of generalized foliated pairs and mention some applications to indicate why such an extension is necessary even for the classical foliated MMP. Time permitting, I will mention some interesting open problems in this field.