Varieties of general type are the higher dimensional analog of Riemann surfaces of genus \(g\geq 2\). If \(X\subset \mathbb P ^N _{\mathbb C}\) is a variety of general type then, by definition, the sections of \(H^0(\omega _X^{\otimes m})\) determine a birational map for all sufficiently big integers \(m>0\).
In these lectures we will explain recent results on the boundedness of varieties of general type that ultimately lead to the construction of a corresponding moduli space (more precisely to the construction of the KSBA proper functor of log canonically polarized log canonical pairs).