  We study the allocation of indivisible goods among groups of agents using
well-known fairness notions such as envy-freeness and proportionality. While
these notions cannot always be satisfied, we provide several bounds on the
optimal relaxations that can be guaranteed. For instance, our bounds imply that
when the number of groups is constant and the $n$ agents are divided into
groups arbitrarily, there exists an allocation that is envy-free up to
$\Theta(\sqrt{n})$ goods, and this bound is tight. Moreover, we show that while
such an allocation can be found efficiently, it is NP-hard to compute an
allocation that is envy-free up to $o(\sqrt{n})$ goods even when a fully
envy-free allocation exists. Our proofs make extensive use of tools from
discrepancy theory.
