  We present a game semantics for Linear Logic, in which formulas denote games
and proofs denote winning strategies. We show that our semantics yields a
categorical model of Linear Logic and prove full completeness for
Multiplicative Linear Logic with the MIX rule: every winning strategy is the
denotation of a unique cut-free proof net. A key role is played by the notion
of {\em history-free} strategy; strong connections are made between
history-free strategies and the Geometry of Interaction. Our semantics
incorporates a natural notion of polarity, leading to a refined treatment of
the additives. We make comparisons with related work by Joyal, Blass et al.
