Algebraic geometry is concerned with algebraic varieties, which can be understood as solution sets of polynomial equations. At the heart of research is the classification of algebraic varieties, and a geometric solution is provided in the form of a moduli space. Roughly speaking, a moduli space is itself an algebro-geometric object whose points represent equivalence classes of algebraic varieties of a fixed type. This talk begins with the moduli space of curves, which parametrizes equivalence classes of complex algebraic curves (i.e. Riemann surfaces) of a fixed genus. This moduli space, like most moduli spaces appearing in algebraic geometry, is not a compact space. A celebrated result of Deligne and Mumford provides a geometric way to compactify this space. The goal of this talk is to discuss recent progress towards compactifying moduli spaces of higher dimensional complex algebraic varieties (e.g. complex algebraic surfaces).