Extreme climate events may be defined as atmospheric or oceanic phenomena that occupy the tails of a dataset's probability density function (PDF), where the magnitude of the event is large but the probability of occurrence is rare. Though these types of events are statistically sparse, it is necessary to understand the distribution of events in the tails, as quantifying the likelihood of climate extremes is an important step in predicting overall climate variability. It has been known for some time that the PDF's of atmospheric phenomena are decidedly non-Gaussian, though the shape of PDF has not been specified explicitly. More recently, it has been shown from observations that many atmospheric variables follow a power law distribution in the tails. This is in agreement with stochastic theory, which asserts that power law distributions should exist in the tails. However, a statistically rigorous study of the resulting power law distributions has not yet been performed. To show the relationship systematically, we examine the PDF tails of dynamically significant atmospheric variables (such as geopotential height and relative vorticity) for evidence of power law behavior. This is achieved by using statistical algorithms that test PDFs for the bounds and magnitude of power law distributions, while estimating the statistical significance of the distribution compared with Gaussianity. Examples of power law distributions in the atmosphere are presented using local time series of atmospheric data, as well as with global maps of power law exponents.