The main objects of study in algebraic geometry are varieties, which are geometric objects locally defined by polynomial equations, and one goal of the subject is to classify all algebraic varieties of a given type.  We approach this problem by constructing parameter spaces, called moduli spaces, whose points correspond to the geometric objects we aim to parameterize.  Depending on the type of variety, there are several different ways to construct a compact moduli space and in this talk we will survey these different moduli spaces and stability conditions (such as GIT stability, K-stability, KSB/KSBA-stability), discuss their relationships, and give several applications.