  We prove a number of results on the determinacy of $\sigma$-projective sets
of reals, i.e., those belonging to the smallest pointclass containing the open
sets and closed under complements, countable unions, and projections. We first
prove the equivalence between $\sigma$-projective determinacy and the
determinacy of certain classes of games of variable length ${<}\omega^2$
(Theorem 2.4). We then give an elementary proof of the determinacy of
$\sigma$-projective sets from optimal large-cardinal hypotheses (Theorem 4.4).
Finally, we show how to generalize the proof to obtain proofs of the
determinacy of $\sigma$-projective games of a given countable length and of
games with payoff in the smallest $\sigma$-algebra containing the projective
sets, from corresponding assumptions (Theorems 5.1 and 5.4).
