  We consider a novel setting where a set of items are matched to the same set
of agents repeatedly over multiple rounds. Each agent gets exactly one item per
round, which brings interesting challenges to finding efficient and/or fair
{\em repeated matchings}. A particular feature of our model is that the value
of an agent for an item in some round depends on the number of rounds in which
the item has been used by the agent in the past. We present a set of positive
and negative results about the efficiency and fairness of repeated matchings.
For example, when items are goods, a variation of the well-studied fairness
notion of envy-freeness up to one good (EF1) can be satisfied under certain
conditions. Furthermore, it is intractable to achieve fairness and
(approximate) efficiency simultaneously, even though they are achievable
separately. For mixed items, which can be goods for some agents and chores for
others, we propose and study a new notion of fairness that we call {\em swap
envy-freeness} (swapEF).

Computer Science and Game Theory;Multiagent Systems