Fractons emerge as charges with reduced mobility in a class of gauge theories. Here, we generalize fractonic theories of U(1) type to what we call (k, n)-fractonic Maxwell theory, which employs symmetric rank-n tensors of k forms (rank-k antisymmetric tensors) as vector potentials.″ The generalization, valid in any spatial dimension d, has two key manifestations. First, the objects with mobility restrictions extend beyond simple charges to higher-order multipoles (dipoles, quadrupoles, etc.) all the way to (n - 1)th-order multipoles, which we call the order-n fracton condition. Second, these fractonic charges themselves are characterized by tensorial densities of (k - 1)-dimensional extended objects. For any (k, n), the theory can be constructed to have a gapless photon modes″ with dispersion omega similar to vertical bar q vertical bar(z), where the integer z can range from 1 to n.