Abstract: I will give a brief introduction to the classical theory of birational geometry, in particular the minimal model program(MMP). Singularities play a crucial role in birational geometry and MMP. They are often measured by numbers called log discrepancies, and the “worst” type of singularities for which MMP works are called Log Canonical, or LC for short. I will then discuss some recent progresses in birational geometry in the LC category, including the finite generation of a log canonical ring and the boundedness on log Fano 3-folds.