The aim of this chapter is to give a comprehensive showing of the main topics concerning combinatorial numbers and polynomials with their generating functions, which have been studied up to the present and with future investigations. For the results of this chapter, we make an effort to use methods of proof with main definitions and notations as elementary as possible for the reader. The topics in this chapter are discussed with the fact that special numbers, special polynomials, and special functions and their generating functions form the basis of not only mathematics but also many applied sciences at almost every stage. In particular, special combinatorial polynomials and numbers containing sums of powers of (inverse) binomial coefficients with their generating functions are given. New results, including their derivative formulas, integral formulas, and finite sums derived from them, are presented by blending and examining them with the previously given results. The relations between these special combinatorial polynomials and their numbers, not only with arithmetic functions but also with other special numbers and special polynomials, are investigated and open problems involving them are also presented. New and known formulas are given by blending known methods, techniques, and known results for the applications of special finite sums containing these combinatorial numbers and polynomials in approximation theory.