  The paper is devoted to the study of Gromov-Hausdorff convergence and
stability of irreversible metric-measure spaces, both in the compact and
noncompact cases. While the compact setting is mostly similar to the reversible
case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case
provides various surprising phenomena. Since the reversibility of noncompact
irreversible spaces might be infinite, it is motivated to introduce a suitable
nondecreasing function that bounds the reversibility of larger and larger
balls. By this approach, we are able to prove satisfactory
convergence/stability results in a suitable -- reversibility depending --
Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by
Finsler manifolds, which serve to construct various model examples by pointing
out genuine differences between the reversible and irreversible settings. We
conclude the paper by proving various geometric and functional inequalities (as
Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on
irreversible structures.
