Exponential families are a large family of probability density functions (PDF) and probability mass functions (PMF) that share certain convenient and important mathematical properties that are necessary for much of traditional inferential theory and methods.  In this lesson, exponential families are defined and illustrated for both the discrete and continuous cases.  An important theorem about the properties of expectations and variances of exponential families is given.  The natural parameter space is defined, and finally, curved exponential families are defined.  All concepts are illustrated using some of the named distribution families of the previous three weeks' lessons.