  We consider a generalization of Hausdorff operator and introduce the notion
of the symbol of such an operator. Using this notion we describe the structure
and investigate important properties (such as invertibility, spectrum, norm,
and compactness) of normal generalized Hausdorff operators on Lebesgue spaces
over $\mathbb{R}^n.$ The examples of Ces\`{a}ro operators are considered.
