The generalizations of the classical orthogonal polynomials satisfying higher-order linear differential equations of a special structure were studied by a number of authors (A. Krall, J. Koekoek, R. Koekoek, H. Bavinck, L. Littlejohn, and several others). The essential requirements were the following. The coefficients of the derivatives must be polynomials of some degree of the independent variable and not dependent on the degree of the polynomials satisfying these differential equations. Such generalizations in the works of the above-mentio ned authors were made for all classical orthogonal polynomials except for the Hermite polynomials. This paper deals with a generalization of the classical Hermite polynomials in the above sense. We construct a differential operator of the infinite order whose eigenfunctions are these polynomials. A number of properties of the generalized Hermite polynomials that are characteristic of classical orthogonal polynomials (orthogonality, generalized Rodrigues’ formula, three-term recurrence relation, generic function) are investigated.