Fermionic statistics include the parity superselection rule, having to do with the global Z_2 (parity) symmetry of fermionic Hamiltonians. In lattice theories, this symmetry can be gauged and made local, with "parity Gauss laws" relating the local fermionic parity on each site with the divergence of parity gauge fields around it. These local constraints allow one to transfer the statistics from the fermionic matter to the gauge field. Lattice gauge theories, whose gauge group contains a normal Z_2 subgroup, offer this possibility without introducing any extra ingredients. I will discuss a transformation we have recently derived, which maps lattice gauge theories with fermionic matter to equivalent theories with hardcore bosonic matter. In the resulting models, the interaction of the matter with the gauge fields is slightly changed, but in a local way. Unlike in the Jordan-Wigner procedure, this mapping gives rise to no nonlocality and can be performed in arbitrary space dimensions. Furthermore, it can be extended to models without any gauge field, by the introduction of an auxiliary, non-dynamical Z_2 field which will absorb the statistics.