In this chapter we will be primarily interested in the study of monomials and polynomials within the framework of quaternion analysis. Monomials and their applications to combinatorics and number theory have become increasingly important for the study of a large number of problems that arise in many different contexts, both from a theoretical and a practical perspective. At the same time, the applications of polynomials to classical and numerical analysis, including approximation theory, statistics, combinatorics, number theory, group representations etc., as well as in physics, including quantum mechanics and statistical physics, and in system theory and signal processing have played a key role in this development and continue to do it today. For example, polynomials are often used in the treatment of problems, mainly in mathematical physics, and also in studies related to differential equations, continued fractions, and numerical stability. In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.