  We introduce generic bosonic models exemplifying that chiral Meissner
currents can persist in insulating phases of matter. We first consider
interacting bosons on a two-leg ladder. The total density sector can be gapped
in a bosonic Mott insulator at odd-integer filling, while the relative density
sector remains superfluid due to interchain hopping. Coupling the relative
density to gauge fields yields a pseudospin Meissner effect. We show that the
same phase arises if the bosons are replaced by spinful fermions confined in
Cooper pairs, and find a dual fermionic Mott insulator with spinon currents. We
prove that by tuning the mean density the Mott insulator with Meissner currents
turns into a low-dimensional bosonic $\nu = \frac{1}{2}$ Laughlin state for
strong enough repulsive interactions across the ladder rungs. We finally
discuss extensions to multileg ladders and bilayers in which spinon superfluids
with Meissner currents become possible. We propose two experimental
realizations, one with ultracold atoms in the setup of Atala et al., Nat. Phys.
\textbf{8}, 588 (2014) and another with Josephson junction arrays. We also
address a Bose-Fermi mixture subject to a magnetic field in connection with the
pseudo-gap phase of high-Tc cuprates.
