  In the Zermelo--Fraenkel set theory with the Axiom of Choice a forcing notion
is "$\kappa$-distributive" if and only if it is "$\kappa$-sequential". We show
that without the Axiom of Choice this equivalence fails, even if we include a
weak form of the Axiom of Choice, the Principle of Dependent Choice for
$\kappa$. Still, the equivalence may still hold along with very strong failures
of the Axiom of Choice, assuming the consistency of large cardinal axioms. We
also prove that while a $\kappa$-distributive forcing notion may violate
Dependent Choice, it must preserve the Axiom of Choice for families of size
$\kappa$. On the other hand, a $\kappa$-sequential can violate the Axiom of
Choice for countable families. We also provide a condition of "quasiproperness"
which is sufficient for the preservation of Dependent Choice, and is also
necessary if the forcing notion is sequential.
