  Fractons emerge as charges with reduced mobility in a new class of gauge
theories. Here, we generalise fractonic theories of $U(1)$ type to what we call
$(k,n)$-fractonic Maxwell theory, which employs symmetric order-$n$ tensors of
$k$-forms (rank-$k$ antisymmetric tensors) as "vector potentials". The
generalisation has two key manifestations. First, the objects with mobility
restrictions extend beyond simple charges to higher order multipoles (dipoles,
quadrupoles, $\ldots$) all the way to $n^\mathrm{th}$-order multipoles. Second,
these fractonic charges themselves are characterized by tensorial densities of
$(k-1)$-dimensional extended objects. The source-free sector exhibits
`photonic' excitations with dispersion $\omega\sim q^n$.
