This is the first part of the review article that focuses on theory and applications of Herglotz-Nevanlinna functions in material sciences. It starts with the definition of scalar-valued Herglotz-Nevanlinna functions and explains in detail the theorems that are pertinent to applications, followed by a short overview of the matrix-valued and operator-valued versions of these functions and the properties that carry over from scalar cases. The theory is complemented by some applications from electromagnetics that are related to the sum rules. More applications of Herglotz-Nevanlinna functions in material sciences can be found in Part II.