The main purpose of this paper is to present Hermite-Chebyshev polynomials and to give some properties of Hermite and Chebyshev polynomials. We derive operational identities, generating functions, and integral representation for power series satisfied by Hermite, Chebyshev, and Hermite-Chebyshev polynomials. Furthermore, for these Hermite-Chebyshev polynomials, we give operational rules with operators, often exploited in the theory of exponential operators. Finally, some definitions of Hermite-Chebyshev polynomials also of two, three and in turn several index are derived and new families of polynomials.