  We introduce the notion of hypersymplectic structure on a Courant algebroid
and we prove the existence of a one-to-one correspondence between
hypersymplectic and hyperk\"ahler structures. This correspondence provides a
simpler way to define a hyperk\"ahler structure on a Courant algebroid. We show
that hypersymplectic structures on Courant algebroids encompass hyperk\"ahler
and hyperk\"ahler structures with torsion on Lie algebroids. In the latter, the
torsion existing at the Lie algebroid level is incorporated in the Courant
structure. Cases of hypersymplectic structures on Courant algebroids which are
doubles of Lie, quasi-Lie and proto-Lie bialgebroids are investigated.
