Heavy-tailed distributions for random variables have been studied extensively, including, for example, in the study of random matrices, in spin glass models in order to bridge the mean-field spin models and the more realistic ones with short-range interactions. In this talk, we consider heavy-tailed distributions that are governed by a power law density. After covering definitions and several fundamental properties, we will discuss a model for random graphs that was developed in an effort to explain the power law behavior observed in many complex networks in reality.