The concepts of convex and concave functions are of critical importance in optimization theory. These concepts arise for both functions of real (that is, continuous) variables and functions of integer variables (that is, discrete) variables, and have been studied extensively by many researchers. In particular, in both real and integer convexity theory many results on the separation of convex sets and necessary conditions for an extremum (so-called Fenchel duality results) have been obtained. However, to the best of our knowledge little or no attention has been given to the study of convexity (and concavity) notions for functions that depend simultaneously on both real and integer variables, a class of functions that we will call "mixed convex/concave functions." In this dissertation we introduce a variety of notions of mixed convexity (and mixed concavity) for both functions and sets. After introducing these various notions, we derive some of their elementary properties and then prove separation and Fenchel duality results for each type of mixed convexity that is introduced.