Almost perfect nonlinear (APN) functions, or differentially 2-uniform functions, are optimally resistant to differential cryptanalysis. As such, the study of APN functions particularly over fields of characteristic two, has important applications in cryptography. We cover the known results on APN functions over both odd and even characteristic fields. We also explore differentially 1-uniform functions, or perfect nonlinear (PN) functions over odd characteristic fields, and almost bent (AB) functions over even characteristic fields. AB functions are a subset of APN functions that have high nonlinearity, making them resistant to linear cryptanalysis. In addition, we explore equivalence classes of APN functions, and what it means for a class of APN functions to be "new", or inequivalent to previously known classes. Finally, we present some new examples of APN functions over the field F53.