It is well-known that a homogeneous polynomial of degree \(k\) admits a harmonic Fischer decomposition. But when dealing with Clifford algebra- valued functions, this decomposition can be refined, since every spherical harmonic can be written as the sum of a so-called inner and an outer spherical monogenic. So every homogeneous Clifford algebra-valued polynomial can be decomposed into so-called Clifford monomials; this is its monogenic decomposition. In this paper we consider operators acting on the space of Clifford algebra-valued polynomials, and in particular the algebra of differential operators with polynomial coefficients. It is clear that the eigenspaces of the Euler operator are the spaces of homogeneous polynomials. The decomposition of polynomials into homogeneous ones leads to the homogeneous decomposition of operators. Moreover, these homogeneous operators are determined by the commutation relation with the Euler operator. It may thus be expected that the further monogenic decomposition of homogeneous polynomials leads toa further decomposition of homogeneous operators into so-called monogenic operators, transforming Clifford monomials into Clifford monomials. Moreover, as these Clifford monomials are in fact the simultaneous eigenfunctions of Euler operator and spherical Dirac operator, it is expected that monogenic operators may be characterized in terms of commutation relations involving these two basic operators. This leads to the notion of monogenic and anti-monogenic operator, and we prove that all operators on polynomials admit such a monogenic decomposition. Finally we establish the monogenic decomposition of the basis differential operators and we give a complete characterization of the \(\text{SO}(m)\)-invariant operators of one vector variables and of several vector variables.