The concept of a function is fundamental to everything we do in mathematics, and also in theoretical computer science. We need some special class of functions to study the size of a set; the growth rate of certain integer-argument functions that help analyze the computational complexity of an algorithm; design of optimal controls; and so on. Our emphasis in this chapter is to introduce certain important class of functions, discuss their properties, and prove related theorems. Some applications to problems in theoretical computer science are also included.