  Auslander and Buchweitz have proved that every finitely generated module over
a Cohen-Macaulay (CM) ring with a dualizing module admits a so-called maximal
CM approximation. In terms of relative homological algebra, this means that
every finitely generated module has a special maximal CM precover. In this
paper, we prove the existence of special maximal CM preenvelopes and, in the
case where the ground ring is henselian, of maximal CM envelopes. We also
characterize the rings over which every finitely generated module has a maximal
CM envelope with the unique lifting property. Finally, we show that cosyzygies
with respect to the class of maximal CM modules must eventually be maximal CM,
and we compute some examples.
