algebraic geometry is the study of shapes defined by polynomial equations called algebraic varieties. one natural approach to study them is to construct a moduli space, which is a space parameterizing such shapes of a given type (e.g. algebraic curves). after surveying this topic, i will focus on the problem of constructing moduli spaces parametrizing fano varieties, which are a class of positively curved complex manifolds that form one of the three main building blocks of varieties in algebraic geometry. while algebraic geometers once considered this problem intractable due to various pathologies that occur, it has recently been solved using k-stability, which is an algebraic definition introduced by differential geometers to characterize when a fano variety admits a kahler-einstein metric.